Part II: Areal Data

Author

Putri Nisrina

7 Spatial neighborhood matrices

Areal or lattice data arise when a study region is divided into a finite number of areas where outcomes are aggregated. Examples include the number of individuals with a certain disease in municipalities of a country, the number of road accidents in provinces, or the average housing prices in districts of a city.

library(spData)
Warning: package 'spData' was built under R version 4.4.2
library(sf)
Warning: package 'sf' was built under R version 4.4.3
Linking to GEOS 3.13.0, GDAL 3.10.1, PROJ 9.5.1; sf_use_s2() is TRUE
library(spdep)
Warning: package 'spdep' was built under R version 4.4.2
library(ggplot2)
map <- st_read(system.file("shapes/columbus.gpkg",
                           package = "spData"), quiet = TRUE)
st_crs(map) <- NA

7.1 Neighbors based on contiguity

Neighbors based on contiguity are constructed by assuming that neighbors of a given area are other areas that share a common boundary.

Here, we use poly2nb() to calculate the neighbors of each region in Columbus based on Queen contiguity.

library(spdep)
nb <- spdep::poly2nb(map, queen = TRUE)
head(nb)
[[1]]
[1] 2 3

[[2]]
[1] 1 3 4

[[3]]
[1] 1 2 4 5

[[4]]
[1] 2 3 5 8

[[5]]
[1]  3  4  6  8  9 11 15 16

[[6]]
[1] 5 9
plot(st_geometry(map), border = "lightgray")
plot.nb(nb, st_geometry(map), add = TRUE)

Map of neighbors based on contiguity.

We can plot the neighbors of a given area by adding a new column in map representing the neighbors of the area.

id <- 20 # area id
map$neighbors <- "other"
map$neighbors[id] <- "area"
map$neighbors[nb[[id]]] <- "neighbors"
ggplot(map) + geom_sf(aes(fill = neighbors)) +
  theme_bw() + scale_fill_manual(values = c("gray30", "gray", "white"))

Map of neighbors of area 20 based on contiguity.

7.2 Neighbors based on k nearest neighbors

We can also consider as neighbors of an area its k nearest neighbors based on the distance separating them.

Neighbors based on 3 nearest neighbors.
# Neighbors based on 3 nearest neighbors
coo <- st_centroid(map)
Warning: st_centroid assumes attributes are constant over geometries
nb <- knn2nb(knearneigh(coo, k=3)) # k number nearest neighbors
plot(st_geometry(map), border = "lightgray")
plot.nb(nb, st_geometry(map), add = TRUE)

Map of neighbors based on 3 nearest neighbors

7.3 Neighbors based on distance

Neigborhood structures can also be defined by considering neighbors areas that are within a given distance.

Neighbors based on distance. The area of interest is shown in black, and its neighbors are shown in gray. The circle’s center represents the centroid of the area of interest, and the circle’s radius represents the distance.
# Neighbors based on distance
nb <- dnearneigh(x= st_centroid(map), d1 = 0, d2 = 0.4)
Warning: st_centroid assumes attributes are constant over geometries
Warning in dnearneigh(x = st_centroid(map), d1 = 0, d2 = 0.4): neighbour object
has 18 sub-graphs
plot(st_geometry(map), border = "lightgray")
plot.nb(nb, st_geometry(map), add = TRUE)

Map of neighbors separated by a distance less than 0.4.

Note that we can also determine an appropriate upper distance to ensure that each area has at least k neighbors. This helps in setting a suitable upper distance bound for a preferred number of neighbors k.

coo <- st_centroid(map)
Warning: st_centroid assumes attributes are constant over geometries
# k is the number nearest neighbors
nb1 <- knn2nb(knearneigh(coo, k=1))
Warning in knn2nb(knearneigh(coo, k = 1)): neighbour object has 13 sub-graphs
dist1 <- nbdists(nb1, coo)
summary(unlist(dist1))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.1276  0.2543  0.3164  0.3291  0.4044  0.6189 

The maximum distance is 0.62 and we can take this value as an upper bound of the distance to ensure each area has at least one neighbor.

7.4 Neighbors of order k based on contiguity

Rook and Queen neighbors of first (dark gray) and second (light gray) order.
library(spdep)
nb <- poly2nb(map, queen = TRUE)
nblags <- spdep::nblag(neighbours = nb, maxlag = 2)
# Neighbors of first order
plot(st_geometry(map), border = "lightgray")
plot.nb(nblags[[1]], st_geometry(map), add = TRUE)

Neighbors of first order.

# Neighbors of second order
plot(st_geometry(map), border = "lightgray")
plot.nb(nblags[[2]], st_geometry(map), add = TRUE)

Neighbors of second order.

# Neighbors of order 1 until order 2
nb <- spdep::poly2nb(map, queen = TRUE)
nblagsc <- spdep::nblag_cumul(nblags)
plot(st_geometry(map), border = "lightgray")
plot.nb(nblagsc, st_geometry(map), add = TRUE)

Neighbors of first order until second order.

7.5 Neighborhood matrices

A spatial neighborhood matrix (W) defines the neighborhood structure across the entire study region, where its elements represent spatial weights. The (i, j)-th element of W, denoted as w??????, expresses the spatial connection between areas i and j. Areas that are closer to i are assigned higher weights than those that are farther away.

Left: Areas of the study region. Right: Spatial weight matrix calculated by assuming neighboring areas share a common boundary, and sum of weights for each area.

Spatial weights matrix based on a binary neighbor list

nb <- poly2nb(map, queen = TRUE)
nbw <- spdep::nb2listw(nb, style = "W")
nbw$weights[1:3]
[[1]]
[1] 0.5 0.5

[[2]]
[1] 0.3333333 0.3333333 0.3333333

[[3]]
[1] 0.25 0.25 0.25 0.25
m1 <- listw2mat(nbw)
lattice::levelplot(
  t(m1),
  scales = list(
    y = list(
      at = c(10, 20, 30, 40),
      labels = c(10, 20, 30, 40)
    )
  )
)

Spatial weights matrix based on a binary neighbor list

Spatial weights matrix based on inverse distance values

coo <- st_centroid(map)
Warning: st_centroid assumes attributes are constant over geometries
nb <- poly2nb(map, queen = TRUE)
dists <- nbdists(nb, coo)
ids <- lapply(dists, function(x){1/x})
nbw <- nb2listw(nb, glist = ids, style = "B")
nbw$weights[1:3]
[[1]]
[1] 1.670235 1.725073

[[2]]
[1] 1.670235 1.404736 2.943309

[[3]]
[1] 1.725073 1.404736 1.782670 1.910694
m2 <- listw2mat(nbw)
lattice::levelplot(
  t(m2),
  scales = list(
    y = list(
      at = c(10, 20, 30, 40),
      labels = c(10, 20, 30, 40)
    )
  )
)

Spatial weights matrix based on inverse distance values

8 Spatial autocorrelation

Spatial autocorrelation describes the extent to which a variable is correlated with itself across space. This concept is closely related to Tobler’s First Law of Geography, which states that “everything is related to everything else, but near things are more related than distant things.”

Examples of configurations of areas showing different types of spatial autocorrelation.
library(spData)
library(sf)
library(mapview)
map <- st_read(system.file("shapes/boston_tracts.gpkg", package = "spData"), quiet = TRUE)
map$vble <- map$MEDV
mapview(map, zcol = "vble")

Median prices of owner-occupied housing in $1000 USD in census tracts of Boston in 1978.

8.1 Global Moran’s I

where n is the number of regions, Y??? is the observed value of the variable of interest in region i, and ?? is the mean of all values. w?????? are the spatial weights that represent the spatial proximity between regions i and j, with w?????? = 0 and i, j = 1, ., n.

When the number of regions is sufficiently large, I follows a normal distribution. We can then assess whether a given spatial pattern significantly deviates from a random pattern by comparing its z-score.

8.2 The moran.test() function

# Neighbors
library(spdep)
nb <- poly2nb(map, queen = TRUE) # queen shares point or border
nbw <- nb2listw(nb, style = "W")
# Global Moran's I
gmoran <- moran.test(map$vble, nbw,
alternative = "greater")
gmoran

    Moran I test under randomisation

data:  map$vble  
weights: nbw    

Moran I statistic standard deviate = 23.35, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
     0.6266753872     -0.0019801980      0.0007248686 
gmoran[["estimate"]][["Moran I statistic"]] # Moran's I
[1] 0.6266754
gmoran[["statistic"]] # z-score
Moran I statistic standard deviate 
                           23.3498 
gmoran[["p.value"]] # p-value
[1] 6.923334e-121

We observe the p-value obtained is lower than the significance level 0.05. Then, we reject the null hypothesis and conclude there is evidence for positive spatial autocorrelation.

The same conclusion is obtained if we use a Monte Carlo approach to assess significance.

gmoranMC <- moran.mc(map$vble, nbw, nsim = 999)
gmoranMC

    Monte-Carlo simulation of Moran I

data:  map$vble 
weights: nbw  
number of simulations + 1: 1000 

statistic = 0.62668, observed rank = 1000, p-value = 0.001
alternative hypothesis: greater
hist(gmoranMC$res)
abline(v=gmoranMC$statistic, col = "red")

Histogram of the Moran’s I values for each of the simulated patterns in the Monte Carlo randomization approach. The red line represents the Moran’s I obtained for the real data.

8.3 Moran’s I scatterplot

This plot displays the observations of each area against its spatially lagged values. The spatially lagged value for a given area is calculated as a weighted average of the neighboring values for that area.

moran.plot(map$vble, nbw)

Moran’s I scatterplot showing the observations plotted against their spatially lagged values.

8.4 Local Moran’s I

There is often interest in providing a local measure of similarity between each area’s value and those of its nearby areas. Local Indicators of Spatial Association (LISA) are designed to indicate the extent of significant spatial clustering of similar values around each observation. A desirable property of LISA is that the sum of all local indicators across regions equals a multiple of the global spatial association indicator.

Typically, the values of the LISAs are mapped to show the locations of areas with comparatively high or low local association with their neighboring areas.

8.5 The localmoran() function

Here, we use the localmoran() function to compute the Local Moran’s I for the housing prices data. We set alternative = "greater", which corresponds to testing H0: no or negative spatial autocorrelation vs. H1: positive spatial autocorrelation.

lmoran <- localmoran(map$vble, nbw, alternative = "greater")
head(lmoran)
             Ii          E.Ii       Var.Ii        Z.Ii Pr(z > E(Ii))
1 -0.3457508492 -5.254157e-04 3.275376e-02 -1.90753363  9.717742e-01
2  0.0175875407 -1.626873e-05 2.045711e-03  0.38921049  3.485602e-01
3  0.0123379633 -6.557001e-07 4.089699e-05  1.92939381  2.684100e-02
4 -0.0001654033 -1.059064e-07 1.331742e-05 -0.04529559  5.180641e-01
5  0.3591628595 -1.427815e-04 7.898947e-03  4.04277384  2.641128e-05
6  0.0545610965 -1.625936e-04 1.357382e-02  0.46970410  3.192832e-01
library(tmap)
Warning: package 'tmap' was built under R version 4.4.2
tmap_mode("plot")
ℹ tmap mode set to "plot".
map$lmI <- lmoran[, "Ii"]        # Local Moran's I
map$lmZ <- lmoran[, "Z.Ii"]      # z-scores
map$lmp <- lmoran[, "Pr(z > E(Ii))"]  # p-values (alternative = "greater")

p1 <- tm_shape(map) +
  tm_polygons(col = "vble", title = "vble", style = "quantile") +
  tm_layout(legend.outside = TRUE)
── tmap v3 code detected ───────────────────────────────────────────────────────
[v3->v4] `tm_polygons()`: instead of `style = "quantile"`, use fill.scale =
`tm_scale_intervals()`.
ℹ Migrate the argument(s) 'style' to 'tm_scale_intervals(<HERE>)'
[v3->v4] `tm_polygons()`: use 'fill' for the fill color of polygons/symbols
(instead of 'col'), and 'col' for the outlines (instead of 'border.col').
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'
p2 <- tm_shape(map) +
  tm_polygons(col = "lmI", title = "Local Moran's I", style = "quantile") +
  tm_layout(legend.outside = TRUE)
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'
p3 <- tm_shape(map) +
  tm_polygons(col = "lmZ", title = "Z-score", breaks = c(-Inf, 1.65, Inf)) +
  tm_layout(legend.outside = TRUE)
[v3->v4] `tm_tm_polygons()`: migrate the argument(s) related to the scale of
the visual variable `fill` namely 'breaks' to fill.scale = tm_scale(<HERE>).
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'
p4 <- tm_shape(map) +
  tm_polygons(col = "lmp", title = "p-value", breaks = c(-Inf, 0.05, Inf)) +
  tm_layout(legend.outside = TRUE)
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'
tmap_arrange(p1, p2, p3, p4)
[scale] tm_polygons:() the data variable assigned to 'fill' contains positive and negative values, so midpoint is set to 0. Set 'midpoint = NA' in 'fill.scale = tm_scale_intervals(<HERE>)' to use all visual values (e.g. colors)
[scale] tm_polygons:() the data variable assigned to 'fill' contains positive and negative values, so midpoint is set to 0. Set 'midpoint = NA' in 'fill.scale = tm_scale_intervals(<HERE>)' to use all visual values (e.g. colors)

Boston housing prices, Local Moran’s I, z-scores, and p-values.

tm_shape(map) +
  tm_polygons(
    col = "lmZ",
    title = "Local Moran's I",
    style = "fixed",
    breaks = c(-Inf, -1.96, 1.96, Inf),
    labels = c("Negative SAC", "No SAC", "Positive SAC"),
    palette = c("blue", "white", "red")
  ) +
  tm_layout(legend.outside = TRUE)
── tmap v3 code detected ───────────────────────────────────────────────────────
[v3->v4] `tm_polygons()`: instead of `style = "fixed"`, use fill.scale =
`tm_scale_intervals()`.
ℹ Migrate the argument(s) 'style', 'breaks', 'palette' (rename to 'values'),
  'labels' to 'tm_scale_intervals(<HERE>)'
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'
Multiple palettes called "blue" found: "kovesi.blue", "tableau.blue". The first one, "kovesi.blue", is returned.

Boston areas showing negative, no, and positive spatial auto correlation according to the local Moran’s I.

8.6 Clusters

Local Moran’s I allows us to identify the following types of clusters: - High-High: areas with high values surrounded by neighbors with high values. - High-Low: areas with high values surrounded by neighbors with low values. - Low-High: areas with low values surrounded by neighbors with high values. - Low-Low: areas with low values surrounded by neighbors with low values.

lmoran <- localmoran(map$vble, nbw, alternative = "two.sided")
head(lmoran)
             Ii          E.Ii       Var.Ii        Z.Ii Pr(z != E(Ii))
1 -0.3457508492 -5.254157e-04 3.275376e-02 -1.90753363   5.645152e-02
2  0.0175875407 -1.626873e-05 2.045711e-03  0.38921049   6.971204e-01
3  0.0123379633 -6.557001e-07 4.089699e-05  1.92939381   5.368199e-02
4 -0.0001654033 -1.059064e-07 1.331742e-05 -0.04529559   9.638717e-01
5  0.3591628595 -1.427815e-04 7.898947e-03  4.04277384   5.282257e-05
6  0.0545610965 -1.625936e-04 1.357382e-02  0.46970410   6.385664e-01
map$lmp <- lmoran[,5] #p-values are in column 5
mp <- moran.plot(as.vector(scale(map$vble)), nbw)

Moran’s I scatterplot showing the scaled values plotted against their spatially lagged values.

head(mp)
            x          wx is_inf labels       dfb.1_         dfb.x        dffit
1 -0.51459745  0.67055821  FALSE      1  0.090884905 -4.681542e-02  0.102233800
2 -0.09055093 -0.19384431  FALSE      2 -0.015161538  1.374250e-03 -0.015223692
3  0.01817895  0.67735383  FALSE      3  0.060002252  1.091857e-03  0.060012186
4  0.00730596 -0.02259476  FALSE      4 -0.004886055 -3.573265e-05 -0.004886186
5  0.26825767  1.33622668   TRUE      5  0.107457903  2.885493e-02  0.111264586
6 -0.28626471 -0.19021998  FALSE      6 -0.003358658  9.624167e-04 -0.003493827
      cov.r       cook.d         hat
1 0.9900174 5.193220e-03 0.002500662
2 1.0055204 1.160840e-04 0.001992521
3 0.9987358 1.797813e-03 0.001976939
4 1.0059201 1.196085e-05 0.001976390
5 0.9831866 6.131141e-03 0.002118784
6 1.0061090 6.115479e-06 0.002138557

We create a variable called quadrant to denote the type of cluster for each area, based on its value, its spatially lagged value, and the corresponding p-value. Specifically: - Areas with quadrant = 1 correspond to High-High clusters. - Areas with quadrant = 2 correspond to Low-Low clusters. - Areas with quadrant = 3 correspond to High-Low clusters. - Areas with quadrant = 4 correspond to Low-High clusters. - Areas with quadrant = 5 are non-significant.

map$quadrant <- NA

# High-High
map[(mp$x >= 0 & mp$wx >= 0) & (map$lmp <= 0.05), "quadrant"] <- 1

# Low-Low
map[(mp$x <= 0 & mp$wx <= 0) & (map$lmp <= 0.05), "quadrant"] <- 2

# High-Low
map[(mp$x >= 0 & mp$wx <= 0) & (map$lmp <= 0.05), "quadrant"] <- 3

# Low-High
map[(mp$x <= 0 & mp$wx >= 0) & (map$lmp <= 0.05), "quadrant"] <- 4

# Non-significant
map[(map$lmp > 0.05), "quadrant"] <- 5
tm_shape(map) +
  tm_fill(
    col = "quadrant",
    title = "",
    breaks = c(1, 2, 3, 4, 5, 6),
    palette = c("red", "blue", "lightpink", "skyblue2", "white"),
    labels = c("High-High", "Low-Low", "High-Low",
               "Low-High", "Non-significant")
  ) +
  tm_legend(text.size = 1) +
  tm_borders(alpha = 0.5) +
  tm_layout(frame = FALSE, title = "Clusters") +
  tm_layout(legend.outside = TRUE)
── tmap v3 code detected ───────────────────────────────────────────────────────
[v3->v4] `tm_tm_polygons()`: migrate the argument(s) related to the scale of
the visual variable `fill` namely 'breaks', 'palette' (rename to 'values'),
'labels' to fill.scale = tm_scale(<HERE>).
[v3->v4] `tm_polygons()`: migrate the argument(s) related to the legend of the
visual variable `fill` namely 'title' to 'fill.legend = tm_legend(<HERE>)'
[v3->v4] `tm_borders()`: use `fill_alpha` instead of `alpha`.
[v3->v4] `tm_layout()`: use `tm_title()` instead of `tm_layout(title = )`
[v3->v4] `tm_legend()`: use 'tm_legend()' inside a layer function, e.g.
'tm_polygons(..., fill.legend = tm_legend())'

High-high, low-low, high-low, and low-high clusters detected in the Boston housing prices data

9 Bayesian spatial models

Bayesian hierarchical models (Banerjee et al., 2004) can be used to analyze areal data that arise when an outcome variable is aggregated into areas that form a partition of the study region.

A commonly used spatial model is the Besag-York-Molli� (BYM) model.

The model includes a spatial random effect (u???) that accounts for the spatial dependence between outcomes, indicating that areas located close to each other may have similar values. An additional unstructured exchangeable component (v???) is included to model uncorrelated random noise.

9.1 Bayesian inference with INLA

Bayesian hierarchical models can be fitted using methods such as Integrated Nested Laplace Approximation (INLA) and Markov Chain Monte Carlo (MCMC). INLA provides a fast, approximate Bayesian inference approach for latent Gaussian models, including generalized linear mixed models and spatial or spatio-temporal models.

# Run R As Administrator
#install.packages("INLA",
#                 repos = c("https://cloud.r-project.org",
#                           INLA = "https://inla.r-inla-download.org/R/stable"),
#                 type = "binary")

9.2 Spatial modeling of housing prices

library(sf)
library(spData)
map <- st_read(system.file("shapes/boston_tracts.GPKG",
                           package = "spData"), quiet = TRUE)
library(mapview)
map$vble <- log(map$MEDV)
mapview(map, zcol = "vble")

Logarithm of housing prices in Boston per census tract from the spData package.

library(GGally)
Registered S3 method overwritten by 'GGally':
  method from   
  +.gg   ggplot2
ggpairs(data = map, columns = c("vble", "CRIM", "RM"))

The relationship between the outcome variable - the logarithm of housing price (VBLE) - and the covariates per capita crime rate (CRIM) and number of rooms (RM).

library(spdep)
library(INLA)
Warning: package 'INLA' was built under R version 4.4.2
Loading required package: Matrix
This is INLA_24.12.11 built 2024-12-11 19:58:26 UTC.
 - See www.r-inla.org/contact-us for how to get help.
 - List available models/likelihoods/etc with inla.list.models()
 - Use inla.doc(<NAME>) to access documentation
nb <-poly2nb(map)
head(nb)
[[1]]
[1]   2   3   6   8 311 313 314 369

[[2]]
[1] 1 3 4 6

[[3]]
[1]   1   2   4   5 369 371 375 376

[[4]]
[1] 2 3 5 6

[[5]]
[1]   3   4   6   7 375 376 411 413 418

[[6]]
[1] 1 2 4 5 7 8
nb2INLA("map.adj", nb)
g <- inla.read.graph(filename = "map.adj")
map$re_u <- 1:nrow(map)
map$re_v <- 1:nrow(map)
formula <- vble ~ CRIM + RM +
  f(re_u, model = "besag", graph = g, scale.model = TRUE) +
  f(re_v, model = "iid")
formula <- vble ~ CRIM + RM + f(re_u, model = "bym2", graph = g)
res <- inla(formula, family = "gaussian", data = map,
            control.predictor = list(compute = TRUE),
            control.compute = list(return.marginals.predictor = TRUE))
res$summary.fixed
                    mean          sd  0.025quant    0.5quant   0.975quant
(Intercept)  1.426854150 0.088534457  1.25310149  1.42687863  1.600466729
CRIM        -0.007842695 0.001292705 -0.01037973 -0.00784233 -0.005307747
RM           0.260317207 0.014043671  0.23277841  0.26031322  0.287878834
                    mode          kld
(Intercept)  1.426878553 2.281143e-10
CRIM        -0.007842332 2.381539e-10
RM           0.260313228 2.294824e-10

We observe an intercept of ??0??? = 1.427 with a 95% credible interval of (1.253, 1.600). The coefficient for crime (CRIM) is ??1 = -0.008 with a 95% credible interval of (-0.010, -0.005), indicating that crime is significantly and negatively associated with housing prices. Meanwhile, the coefficient for number of rooms (RM) is ??2 = 0.260 with a 95% credible interval of (0.233, 0.288), suggesting that the number of rooms is significantly and positively associated with housing prices. Overall, these results imply that both crime rate and number of rooms play important roles in explaining the spatial variation of housing prices.

summary(res$summary.fitted.values)
      mean             sd             0.025quant       0.5quant    
 Min.   :1.610   Min.   :0.009883   Min.   :1.589   Min.   :1.610  
 1st Qu.:2.835   1st Qu.:0.009911   1st Qu.:2.813   1st Qu.:2.835  
 Median :3.054   Median :0.009920   Median :3.032   Median :3.054  
 Mean   :3.035   Mean   :0.009931   Mean   :3.013   Mean   :3.035  
 3rd Qu.:3.219   3rd Qu.:0.009930   3rd Qu.:3.198   3rd Qu.:3.219  
 Max.   :3.912   Max.   :0.011217   Max.   :3.891   Max.   :3.912  
   0.975quant         mode      
 Min.   :1.633   Min.   :1.610  
 1st Qu.:2.857   1st Qu.:2.835  
 Median :3.075   Median :3.054  
 Mean   :3.056   Mean   :3.035  
 3rd Qu.:3.241   3rd Qu.:3.219  
 Max.   :3.934   Max.   :3.912  

We can create variables with the posterior mean (PM) and lower (LL) and upper (UL) limits of 95% credible intervals.

# Posterior mean and 95% CI
map$PM <- res$summary.fitted.values[, "mean"]
map$LL <- res$summary.fitted.values[, "0.025quant"]
map$UL <- res$summary.fitted.values[, "0.975quant"]
# Common legend
at <- seq(
  min(c(map$PM, map$LL, map$UL)),
  max(c(map$PM, map$LL, map$UL)),
  length.out = 8
)

# Popup table
popuptable <- leafpop::popupTable(
  dplyr::mutate_if(map, is.numeric, round, digits = 2),
  zcol = c("TOWN", "vble", "CRIM", "RM", "PM", "LL", "UL"),
  row.numbers = FALSE,
  feature.id = FALSE
)

# Map visualizations
m1 <- mapview(
  map,
  zcol = "PM",
  map.types = "CartoDB.Positron",
  at = at,
  popup = popuptable
)

m2 <- mapview(
  map,
  zcol = "LL",
  map.types = "CartoDB.Positron",
  at = at,
  popup = popuptable
)

m3 <- mapview(
  map,
  zcol = "UL",
  map.types = "CartoDB.Positron",
  at = at,
  popup = popuptable
)
library(leafsync)
Warning: package 'leafsync' was built under R version 4.4.2
m <- leafsync::sync(m1,m2,m3,ncol=3)
m

Posterior mean of the logarithm of housing prices (left), along with the lower (center) and upper (right) limits of the 95% credible intervals.

# Transformation of the marginals using inla.tmarginal()
# Example: transformation for the first area
# inla.tmarginal(function(x) exp(x), res$marginals.fitted.values[[1]])

# Transform all marginals
marginals <- lapply(
  res$marginals.fitted.values,
  FUN = function(marg) {
    inla.tmarginal(function(x) exp(x), marg)
  }
)

# Obtain summaries of the transformed marginals using inla.zmarginal()
marginals_summaries <- lapply(
  marginals,
  FUN = function(marg) {
    inla.zmarginal(marg)
  }
)
Mean            17.8214 
Stdev           0.176978 
Quantile  0.025 17.4708 
Quantile  0.25  17.7288 
Quantile  0.5   17.8116 
Quantile  0.75  17.9004 
Quantile  0.975 18.2401 
Mean            21.7029 
Stdev           0.213683 
Quantile  0.025 21.2434 
Quantile  0.25  21.597 
Quantile  0.5   21.7005 
Quantile  0.75  21.805 
Quantile  0.975 22.1753 
Mean            22.6976 
Stdev           0.222831 
Quantile  0.025 22.2102 
Quantile  0.25  22.5884 
Quantile  0.5   22.6974 
Quantile  0.75  22.8058 
Quantile  0.975 23.1818 
Mean            22.5978 
Stdev           0.222391 
Quantile  0.025 22.1114 
Quantile  0.25  22.489 
Quantile  0.5   22.5975 
Quantile  0.75  22.7056 
Quantile  0.975 23.0814 
Mean            25.0053 
Stdev           0.245537 
Quantile  0.025 24.4803 
Quantile  0.25  24.8829 
Quantile  0.5   25.0018 
Quantile  0.75  25.1223 
Quantile  0.975 25.5507 
Mean            19.9089 
Stdev           0.196105 
Quantile  0.025 19.4969 
Quantile  0.25  19.81 
Quantile  0.5   19.9042 
Quantile  0.75  20.0009 
Quantile  0.975 20.3517 
Mean            20.7922 
Stdev           0.204449 
Quantile  0.025 20.3364 
Quantile  0.25  20.6936 
Quantile  0.5   20.7943 
Quantile  0.75  20.893 
Quantile  0.975 21.2279 
Mean            16.8141 
Stdev           0.165825 
Quantile  0.025 16.4762 
Quantile  0.25  16.7287 
Quantile  0.5   16.8074 
Quantile  0.75  16.8901 
Quantile  0.975 17.198 
Mean            22.0037 
Stdev           0.249736 
Quantile  0.025 21.6288 
Quantile  0.25  21.8587 
Quantile  0.5   21.9585 
Quantile  0.75  22.0884 
Quantile  0.975 22.6811 
Mean            27.4507 
Stdev           0.274942 
Quantile  0.025 26.7834 
Quantile  0.25  27.3296 
Quantile  0.5   27.4696 
Quantile  0.75  27.5965 
Quantile  0.975 27.9755 
Mean            21.9174 
Stdev           0.216892 
Quantile  0.025 21.4736 
Quantile  0.25  21.8062 
Quantile  0.5   21.909 
Quantile  0.75  22.0167 
Quantile  0.975 22.4182 
Mean            23.0788 
Stdev           0.227902 
Quantile  0.025 22.5524 
Quantile  0.25  22.9725 
Quantile  0.5   23.0863 
Quantile  0.75  23.1946 
Quantile  0.975 23.5449 
Mean            49.864 
Stdev           0.512281 
Quantile  0.025 48.5665 
Quantile  0.25  49.6523 
Quantile  0.5   49.9176 
Quantile  0.75  50.146 
Quantile  0.975 50.7714 
Mean            49.9376 
Stdev           0.495119 
Quantile  0.025 48.7715 
Quantile  0.25  49.7115 
Quantile  0.5   49.9607 
Quantile  0.75  50.1935 
Quantile  0.975 50.9248 
Mean            49.9617 
Stdev           0.492362 
Quantile  0.025 48.8356 
Quantile  0.25  49.7299 
Quantile  0.5   49.9749 
Quantile  0.75  50.21 
Quantile  0.975 50.9807 
Mean            49.9923 
Stdev           0.49268 
Quantile  0.025 48.9105 
Quantile  0.25  49.7523 
Quantile  0.5   49.9928 
Quantile  0.75  50.2318 
Quantile  0.975 51.0597 
Mean            49.8138 
Stdev           0.532235 
Quantile  0.025 48.4166 
Quantile  0.25  49.6093 
Quantile  0.5   49.8887 
Quantile  0.75  50.116 
Quantile  0.975 50.6855 
Mean            13.8033 
Stdev           0.136253 
Quantile  0.025 13.5125 
Quantile  0.25  13.7355 
Quantile  0.5   13.8012 
Quantile  0.75  13.8679 
Quantile  0.975 14.1068 
Mean            13.8043 
Stdev           0.136216 
Quantile  0.025 13.5151 
Quantile  0.25  13.7363 
Quantile  0.5   13.8018 
Quantile  0.75  13.8686 
Quantile  0.975 14.1092 
Mean            15.0158 
Stdev           0.149123 
Quantile  0.025 14.7169 
Quantile  0.25  14.9384 
Quantile  0.5   15.0085 
Quantile  0.75  15.0829 
Quantile  0.975 15.3657 
Mean            13.8993 
Stdev           0.136983 
Quantile  0.025 13.6008 
Quantile  0.25  13.8322 
Quantile  0.5   13.8989 
Quantile  0.75  13.9655 
Quantile  0.975 14.1984 
Mean            13.3036 
Stdev           0.130981 
Quantile  0.025 13.0248 
Quantile  0.25  13.2382 
Quantile  0.5   13.3014 
Quantile  0.75  13.3657 
Quantile  0.975 13.5959 
Mean            13.1065 
Stdev           0.129069 
Quantile  0.025 12.8363 
Quantile  0.25  13.0412 
Quantile  0.5   13.1031 
Quantile  0.75  13.1668 
Quantile  0.975 13.3988 
Mean            10.2029 
Stdev           0.100589 
Quantile  0.025 9.98885 
Quantile  0.25  10.1527 
Quantile  0.5   10.2011 
Quantile  0.75  10.2505 
Quantile  0.975 10.4275 
Mean            10.3978 
Stdev           0.102377 
Quantile  0.025 10.172 
Quantile  0.25  10.348 
Quantile  0.5   10.3981 
Quantile  0.75  10.4477 
Quantile  0.975 10.6186 
Mean            10.9144 
Stdev           0.108686 
Quantile  0.025 10.7012 
Quantile  0.25  10.8573 
Quantile  0.5   10.9079 
Quantile  0.75  10.9625 
Quantile  0.975 11.1734 
Mean            11.2989 
Stdev           0.111196 
Quantile  0.025 11.0558 
Quantile  0.25  11.2445 
Quantile  0.5   11.2988 
Quantile  0.75  11.3528 
Quantile  0.975 11.5408 
Mean            12.2996 
Stdev           0.120901 
Quantile  0.025 12.0366 
Quantile  0.25  12.2402 
Quantile  0.5   12.2991 
Quantile  0.75  12.3581 
Quantile  0.975 12.5638 
Mean            8.79668 
Stdev           0.0865491 
Quantile  0.025 8.60373 
Quantile  0.25  8.755 
Quantile  0.5   8.79758 
Quantile  0.75  8.83936 
Quantile  0.975 8.98112 
Mean            7.20689 
Stdev           0.0715166 
Quantile  0.025 7.06239 
Quantile  0.25  7.16996 
Quantile  0.5   7.20365 
Quantile  0.75  7.23927 
Quantile  0.975 7.3737 
Mean            10.4887 
Stdev           0.103691 
Quantile  0.025 10.247 
Quantile  0.25  10.4408 
Quantile  0.5   10.4928 
Quantile  0.75  10.5419 
Quantile  0.975 10.6982 
Mean            7.40513 
Stdev           0.0732461 
Quantile  0.025 7.25406 
Quantile  0.25  7.36779 
Quantile  0.5   7.40261 
Quantile  0.75  7.43888 
Quantile  0.975 7.5732 
Mean            10.1984 
Stdev           0.1003 
Quantile  0.025 9.97809 
Quantile  0.25  10.1495 
Quantile  0.5   10.1985 
Quantile  0.75  10.2472 
Quantile  0.975 10.4155 
Mean            11.4992 
Stdev           0.112892 
Quantile  0.025 11.2529 
Quantile  0.25  11.4438 
Quantile  0.5   11.4989 
Quantile  0.75  11.5539 
Quantile  0.975 11.7452 
Mean            15.1009 
Stdev           0.148368 
Quantile  0.025 14.7801 
Quantile  0.25  15.0276 
Quantile  0.5   15.0997 
Quantile  0.75  15.1722 
Quantile  0.975 15.4271 
Mean            23.1715 
Stdev           0.229407 
Quantile  0.025 22.6318 
Quantile  0.25  23.0665 
Quantile  0.5   23.182 
Quantile  0.75  23.29 
Quantile  0.975 23.6294 
Mean            9.7048 
Stdev           0.0954115 
Quantile  0.025 9.50508 
Quantile  0.25  9.65651 
Quantile  0.5   9.70231 
Quantile  0.75  9.74949 
Quantile  0.975 9.92081 
Mean            13.7925 
Stdev           0.135886 
Quantile  0.025 13.486 
Quantile  0.25  13.7277 
Quantile  0.5   13.7948 
Quantile  0.75  13.8601 
Quantile  0.975 14.0785 
Mean            12.6955 
Stdev           0.125064 
Quantile  0.025 12.4171 
Quantile  0.25  12.6352 
Quantile  0.5   12.6967 
Quantile  0.75  12.757 
Quantile  0.975 12.9625 
Mean            13.0993 
Stdev           0.129006 
Quantile  0.025 12.8181 
Quantile  0.25  13.0361 
Quantile  0.5   13.0989 
Quantile  0.75  13.1617 
Quantile  0.975 13.3808 
Mean            12.4957 
Stdev           0.123008 
Quantile  0.025 12.2222 
Quantile  0.25  12.4364 
Quantile  0.5   12.4968 
Quantile  0.75  12.5562 
Quantile  0.975 12.7586 
Mean            8.50453 
Stdev           0.0838291 
Quantile  0.025 8.32954 
Quantile  0.25  8.46208 
Quantile  0.5   8.5022 
Quantile  0.75  8.54362 
Quantile  0.975 8.69486 
Mean            5.00484 
Stdev           0.0495816 
Quantile  0.025 4.90478 
Quantile  0.25  4.97919 
Quantile  0.5   5.00257 
Quantile  0.75  5.0273 
Quantile  0.975 5.12055 
Mean            6.30237 
Stdev           0.0621336 
Quantile  0.025 6.17113 
Quantile  0.25  6.27119 
Quantile  0.5   6.30105 
Quantile  0.75  6.33161 
Quantile  0.975 6.44204 
Mean            5.60803 
Stdev           0.0559972 
Quantile  0.025 5.49912 
Quantile  0.25  5.57844 
Quantile  0.5   5.6044 
Quantile  0.75  5.63258 
Quantile  0.975 5.74226 
Mean            7.20592 
Stdev           0.0712434 
Quantile  0.025 7.06049 
Quantile  0.25  7.16932 
Quantile  0.5   7.20309 
Quantile  0.75  7.23852 
Quantile  0.975 7.3707 
Mean            12.0942 
Stdev           0.119042 
Quantile  0.025 11.8269 
Quantile  0.25  12.0372 
Quantile  0.5   12.0959 
Quantile  0.75  12.1532 
Quantile  0.975 12.3459 
Mean            8.29911 
Stdev           0.0816742 
Quantile  0.025 8.1204 
Quantile  0.25  8.25918 
Quantile  0.5   8.29904 
Quantile  0.75  8.33873 
Quantile  0.975 8.47663 
Mean            8.49486 
Stdev           0.083617 
Quantile  0.025 8.30561 
Quantile  0.25  8.45511 
Quantile  0.5   8.49653 
Quantile  0.75  8.53665 
Quantile  0.975 8.67005 
Mean            5.00366 
Stdev           0.0494342 
Quantile  0.025 4.90201 
Quantile  0.25  4.97838 
Quantile  0.5   5.00188 
Quantile  0.75  5.02641 
Quantile  0.975 5.11734 
Mean            11.9039 
Stdev           0.117557 
Quantile  0.025 11.6545 
Quantile  0.25  11.8451 
Quantile  0.5   11.9016 
Quantile  0.75  11.9593 
Quantile  0.975 12.1673 
Mean            27.8914 
Stdev           0.27403 
Quantile  0.025 27.2834 
Quantile  0.25  27.7588 
Quantile  0.5   27.8935 
Quantile  0.75  28.0261 
Quantile  0.975 28.4782 
Mean            17.213 
Stdev           0.170151 
Quantile  0.025 16.8638 
Quantile  0.25  17.1259 
Quantile  0.5   17.2067 
Quantile  0.75  17.2911 
Quantile  0.975 17.6049 
Mean            27.4996 
Stdev           0.27049 
Quantile  0.025 26.9116 
Quantile  0.25  27.3666 
Quantile  0.5   27.4983 
Quantile  0.75  27.6301 
Quantile  0.975 28.0913 
Mean            15.0071 
Stdev           0.148068 
Quantile  0.025 14.6967 
Quantile  0.25  14.9325 
Quantile  0.5   15.0034 
Quantile  0.75  15.0764 
Quantile  0.975 15.3421 
Mean            17.235 
Stdev           0.174531 
Quantile  0.025 16.9119 
Quantile  0.25  17.1405 
Quantile  0.5   17.2196 
Quantile  0.75  17.3083 
Quantile  0.975 17.6666 
Mean            17.8788 
Stdev           0.177112 
Quantile  0.025 17.4632 
Quantile  0.25  17.7976 
Quantile  0.5   17.8866 
Quantile  0.75  17.9701 
Quantile  0.975 18.2337 
Mean            16.2783 
Stdev           0.161604 
Quantile  0.025 15.8958 
Quantile  0.25  16.2049 
Quantile  0.5   16.2864 
Quantile  0.75  16.3622 
Quantile  0.975 16.5984 
Mean            6.99166 
Stdev           0.0691612 
Quantile  0.025 6.82933 
Quantile  0.25  6.95993 
Quantile  0.5   6.99472 
Quantile  0.75  7.02735 
Quantile  0.975 7.13014 
Mean            7.20559 
Stdev           0.0712911 
Quantile  0.025 7.05952 
Quantile  0.25  7.16908 
Quantile  0.5   7.20289 
Quantile  0.75  7.23828 
Quantile  0.975 7.37003 
Mean            7.51283 
Stdev           0.0755261 
Quantile  0.025 7.3692 
Quantile  0.25  7.47248 
Quantile  0.5   7.50711 
Quantile  0.75  7.54528 
Quantile  0.975 7.6966 
Mean            10.4088 
Stdev           0.103071 
Quantile  0.025 10.1988 
Quantile  0.25  10.3558 
Quantile  0.5   10.4046 
Quantile  0.75  10.4558 
Quantile  0.975 10.6476 
Mean            8.79058 
Stdev           0.0869942 
Quantile  0.025 8.58782 
Quantile  0.25  8.75041 
Quantile  0.5   8.79399 
Quantile  0.75  8.83513 
Quantile  0.975 8.96648 
Mean            8.40673 
Stdev           0.0833143 
Quantile  0.025 8.23633 
Quantile  0.25  8.36404 
Quantile  0.5   8.40349 
Quantile  0.75  8.44483 
Quantile  0.975 8.59924 
Mean            16.6866 
Stdev           0.164805 
Quantile  0.025 16.3087 
Quantile  0.25  16.6093 
Quantile  0.5   16.6912 
Quantile  0.75  16.7697 
Quantile  0.975 17.0268 
Mean            14.2163 
Stdev           0.141215 
Quantile  0.025 13.9353 
Quantile  0.25  14.1426 
Quantile  0.5   14.2088 
Quantile  0.75  14.2795 
Quantile  0.975 14.5494 
Mean            20.7735 
Stdev           0.205927 
Quantile  0.025 20.2877 
Quantile  0.25  20.6795 
Quantile  0.5   20.7833 
Quantile  0.75  20.8801 
Quantile  0.975 21.1831 
Mean            13.4091 
Stdev           0.13236 
Quantile  0.025 13.136 
Quantile  0.25  13.3416 
Quantile  0.5   13.4046 
Quantile  0.75  13.4703 
Quantile  0.975 13.7126 
Mean            11.7001 
Stdev           0.114943 
Quantile  0.025 11.4507 
Quantile  0.25  11.6435 
Quantile  0.5   11.6994 
Quantile  0.75  11.7555 
Quantile  0.975 11.9519 
Mean            8.31729 
Stdev           0.0842312 
Quantile  0.025 8.16189 
Quantile  0.25  8.27157 
Quantile  0.5   8.30968 
Quantile  0.75  8.35256 
Quantile  0.975 8.52596 
Mean            10.2027 
Stdev           0.100448 
Quantile  0.025 9.98886 
Quantile  0.25  10.1526 
Quantile  0.5   10.2011 
Quantile  0.75  10.2504 
Quantile  0.975 10.4269 
Mean            10.8964 
Stdev           0.107263 
Quantile  0.025 10.6579 
Quantile  0.25  10.8446 
Quantile  0.5   10.8973 
Quantile  0.75  10.9491 
Quantile  0.975 11.1257 
Mean            11.002 
Stdev           0.108215 
Quantile  0.025 10.7702 
Quantile  0.25  10.9482 
Quantile  0.5   11.0006 
Quantile  0.75  11.0536 
Quantile  0.975 11.2421 
Mean            9.50502 
Stdev           0.0935241 
Quantile  0.025 9.30978 
Quantile  0.25  9.45762 
Quantile  0.5   9.50244 
Quantile  0.75  9.5487 
Quantile  0.975 9.71728 
Mean            14.4952 
Stdev           0.14242 
Quantile  0.025 14.1786 
Quantile  0.25  14.4263 
Quantile  0.5   14.4964 
Quantile  0.75  14.5653 
Quantile  0.975 14.7996 
Mean            14.0993 
Stdev           0.138584 
Quantile  0.025 13.7973 
Quantile  0.25  14.0313 
Quantile  0.5   14.0988 
Quantile  0.75  14.1664 
Quantile  0.975 14.4017 
Mean            16.0968 
Stdev           0.158313 
Quantile  0.025 15.7483 
Quantile  0.25  16.0198 
Quantile  0.5   16.0973 
Quantile  0.75  16.1741 
Quantile  0.975 16.4387 
Mean            14.303 
Stdev           0.140687 
Quantile  0.025 14.0022 
Quantile  0.25  14.233 
Quantile  0.5   14.301 
Quantile  0.75  14.37 
Quantile  0.975 14.6156 
Mean            11.7023 
Stdev           0.115098 
Quantile  0.025 11.4558 
Quantile  0.25  11.645 
Quantile  0.5   11.7007 
Quantile  0.75  11.7571 
Quantile  0.975 11.9577 
Mean            13.3967 
Stdev           0.131604 
Quantile  0.025 13.106 
Quantile  0.25  13.3328 
Quantile  0.5   13.3974 
Quantile  0.75  13.4612 
Quantile  0.975 13.6799 
Mean            9.60468 
Stdev           0.0946953 
Quantile  0.025 9.40632 
Quantile  0.25  9.55686 
Quantile  0.5   9.60223 
Quantile  0.75  9.64895 
Quantile  0.975 9.81907 
Mean            8.70338 
Stdev           0.0857473 
Quantile  0.025 8.52243 
Quantile  0.25  8.6603 
Quantile  0.5   8.70152 
Quantile  0.75  8.74373 
Quantile  0.975 8.89626 
Mean            8.40506 
Stdev           0.0828987 
Quantile  0.025 8.23291 
Quantile  0.25  8.36293 
Quantile  0.5   8.40252 
Quantile  0.75  8.44354 
Quantile  0.975 8.59412 
Mean            12.7919 
Stdev           0.125794 
Quantile  0.025 12.5066 
Quantile  0.25  12.7321 
Quantile  0.5   12.7945 
Quantile  0.75  12.8549 
Quantile  0.975 13.0548 
Mean            10.5052 
Stdev           0.103718 
Quantile  0.025 10.288 
Quantile  0.25  10.4528 
Quantile  0.5   10.5025 
Quantile  0.75  10.5536 
Quantile  0.975 10.7402 
Mean            17.0954 
Stdev           0.167945 
Quantile  0.025 16.7239 
Quantile  0.25  17.014 
Quantile  0.5   17.0964 
Quantile  0.75  17.1778 
Quantile  0.975 17.4562 
Mean            18.3883 
Stdev           0.181146 
Quantile  0.025 17.9774 
Quantile  0.25  18.3024 
Quantile  0.5   18.3921 
Quantile  0.75  18.4789 
Quantile  0.975 18.7669 
Mean            15.3995 
Stdev           0.151368 
Quantile  0.025 15.0702 
Quantile  0.25  15.3252 
Quantile  0.5   15.3989 
Quantile  0.75  15.4727 
Quantile  0.975 15.7303 
Mean            10.7998 
Stdev           0.106157 
Quantile  0.025 10.569 
Quantile  0.25  10.7476 
Quantile  0.5   10.7993 
Quantile  0.75  10.8511 
Quantile  0.975 11.032 
Mean            11.8034 
Stdev           0.116075 
Quantile  0.025 11.5566 
Quantile  0.25  11.7453 
Quantile  0.5   11.8014 
Quantile  0.75  11.8584 
Quantile  0.975 12.0627 
Mean            14.8985 
Stdev           0.146346 
Quantile  0.025 14.5786 
Quantile  0.25  14.8269 
Quantile  0.5   14.8983 
Quantile  0.75  14.9696 
Quantile  0.975 15.2168 
Mean            12.6062 
Stdev           0.124211 
Quantile  0.025 12.346 
Quantile  0.25  12.5434 
Quantile  0.5   12.6029 
Quantile  0.75  12.6643 
Quantile  0.975 12.8874 
Mean            14.0958 
Stdev           0.138745 
Quantile  0.025 13.788 
Quantile  0.25  14.0287 
Quantile  0.5   14.0967 
Quantile  0.75  14.1638 
Quantile  0.975 14.3931 
Mean            13.0024 
Stdev           0.127801 
Quantile  0.025 12.7285 
Quantile  0.25  12.9388 
Quantile  0.5   13.0007 
Quantile  0.75  13.0633 
Quantile  0.975 13.2857 
Mean            13.4079 
Stdev           0.132142 
Quantile  0.025 13.1331 
Quantile  0.25  13.3407 
Quantile  0.5   13.4039 
Quantile  0.75  13.4693 
Quantile  0.975 13.7089 
Mean            15.2012 
Stdev           0.149452 
Quantile  0.025 14.8786 
Quantile  0.25  15.1273 
Quantile  0.5   15.1999 
Quantile  0.75  15.2729 
Quantile  0.975 15.5303 
Mean            16.0996 
Stdev           0.158155 
Quantile  0.025 15.7557 
Quantile  0.25  16.0218 
Quantile  0.5   16.0989 
Quantile  0.75  16.1761 
Quantile  0.975 16.4454 
Mean            17.804 
Stdev           0.175359 
Quantile  0.025 17.4293 
Quantile  0.25  17.7167 
Quantile  0.5   17.8014 
Quantile  0.75  17.8873 
Quantile  0.975 18.1941 
Mean            14.9005 
Stdev           0.146392 
Quantile  0.025 14.5835 
Quantile  0.25  14.8283 
Quantile  0.5   14.8995 
Quantile  0.75  14.971 
Quantile  0.975 15.2219 
Mean            14.106 
Stdev           0.138922 
Quantile  0.025 13.8136 
Quantile  0.25  14.036 
Quantile  0.5   14.1028 
Quantile  0.75  14.1712 
Quantile  0.975 14.4192 
Mean            12.7089 
Stdev           0.125393 
Quantile  0.025 12.4504 
Quantile  0.25  12.6448 
Quantile  0.5   12.7045 
Quantile  0.75  12.7667 
Quantile  0.975 12.9966 
Mean            13.5061 
Stdev           0.133214 
Quantile  0.025 13.2264 
Quantile  0.25  13.439 
Quantile  0.5   13.5029 
Quantile  0.75  13.5685 
Quantile  0.975 13.8071 
Mean            14.9126 
Stdev           0.147393 
Quantile  0.025 14.6123 
Quantile  0.25  14.8368 
Quantile  0.5   14.9066 
Quantile  0.75  14.98 
Quantile  0.975 15.254 
Mean            19.9912 
Stdev           0.196857 
Quantile  0.025 19.5504 
Quantile  0.25  19.8968 
Quantile  0.5   19.9937 
Quantile  0.75  20.0886 
Quantile  0.975 20.4088 
Mean            16.4073 
Stdev           0.161332 
Quantile  0.025 16.0684 
Quantile  0.25  16.3259 
Quantile  0.5   16.4034 
Quantile  0.75  16.4831 
Quantile  0.975 16.7715 
Mean            17.7066 
Stdev           0.174317 
Quantile  0.025 17.3382 
Quantile  0.25  17.619 
Quantile  0.5   17.7029 
Quantile  0.75  17.7887 
Quantile  0.975 18.0981 
Mean            19.5023 
Stdev           0.192005 
Quantile  0.025 19.0889 
Quantile  0.25  19.4073 
Quantile  0.5   19.5003 
Quantile  0.75  19.5942 
Quantile  0.975 19.9263 
Mean            20.2049 
Stdev           0.198651 
Quantile  0.025 19.7812 
Quantile  0.25  20.1058 
Quantile  0.5   20.2018 
Quantile  0.75  20.2994 
Quantile  0.975 20.6473 
Mean            21.3947 
Stdev           0.2103 
Quantile  0.025 20.9301 
Quantile  0.25  21.2926 
Quantile  0.5   21.3958 
Quantile  0.75  21.4977 
Quantile  0.975 21.8472 
Mean            19.8911 
Stdev           0.195292 
Quantile  0.025 19.4538 
Quantile  0.25  19.7973 
Quantile  0.5   19.8937 
Quantile  0.75  19.9879 
Quantile  0.975 20.305 
Mean            18.9913 
Stdev           0.186788 
Quantile  0.025 18.5726 
Quantile  0.25  18.9017 
Quantile  0.5   18.9939 
Quantile  0.75  19.0839 
Quantile  0.975 19.3869 
Mean            19.08 
Stdev           0.187685 
Quantile  0.025 18.6435 
Quantile  0.25  18.9928 
Quantile  0.5   19.0872 
Quantile  0.75  19.1763 
Quantile  0.975 19.46 
Mean            19.0971 
Stdev           0.187705 
Quantile  0.025 18.6851 
Quantile  0.25  19.0055 
Quantile  0.5   19.0973 
Quantile  0.75  19.1885 
Quantile  0.975 19.5037 
Mean            20.0942 
Stdev           0.197916 
Quantile  0.025 19.6555 
Quantile  0.25  19.9985 
Quantile  0.5   20.0955 
Quantile  0.75  20.1912 
Quantile  0.975 20.5188 
Mean            19.9072 
Stdev           0.195852 
Quantile  0.025 19.4931 
Quantile  0.25  19.8089 
Quantile  0.5   19.9032 
Quantile  0.75  19.9996 
Quantile  0.975 20.3468 
Mean            19.6052 
Stdev           0.193139 
Quantile  0.025 19.1939 
Quantile  0.25  19.5089 
Quantile  0.5   19.602 
Quantile  0.75  19.6968 
Quantile  0.975 20.0361 
Mean            23.2027 
Stdev           0.227873 
Quantile  0.025 22.7121 
Quantile  0.25  23.0897 
Quantile  0.5   23.2003 
Quantile  0.75  23.3119 
Quantile  0.975 23.7057 
Mean            29.7858 
Stdev           0.292635 
Quantile  0.025 29.1291 
Quantile  0.25  29.6455 
Quantile  0.5   29.7901 
Quantile  0.75  29.9311 
Quantile  0.975 30.4047 
Mean            13.8018 
Stdev           0.13589 
Quantile  0.025 13.5095 
Quantile  0.25  13.7345 
Quantile  0.5   13.8003 
Quantile  0.75  13.8668 
Quantile  0.975 14.1022 
Mean            13.309 
Stdev           0.131443 
Quantile  0.025 13.0376 
Quantile  0.25  13.242 
Quantile  0.5   13.3046 
Quantile  0.75  13.3697 
Quantile  0.975 13.6102 
Mean            16.7097 
Stdev           0.164775 
Quantile  0.025 16.367 
Quantile  0.25  16.6261 
Quantile  0.5   16.7048 
Quantile  0.75  16.7863 
Quantile  0.975 17.085 
Mean            11.9964 
Stdev           0.118007 
Quantile  0.025 11.7347 
Quantile  0.25  11.9393 
Quantile  0.5   11.9972 
Quantile  0.75  12.0543 
Quantile  0.975 12.2493 
Mean            14.6046 
Stdev           0.143916 
Quantile  0.025 14.2993 
Quantile  0.25  14.5326 
Quantile  0.5   14.6019 
Quantile  0.75  14.6726 
Quantile  0.975 14.9268 
Mean            21.3866 
Stdev           0.210538 
Quantile  0.025 20.9094 
Quantile  0.25  21.2866 
Quantile  0.5   21.391 
Quantile  0.75  21.4919 
Quantile  0.975 21.8269 
Mean            23.0107 
Stdev           0.22642 
Quantile  0.025 22.5356 
Quantile  0.25  22.8963 
Quantile  0.5   23.005 
Quantile  0.75  23.1168 
Quantile  0.975 23.5223 
Mean            23.7056 
Stdev           0.2335 
Quantile  0.025 23.2072 
Quantile  0.25  23.5893 
Quantile  0.5   23.702 
Quantile  0.75  23.8165 
Quantile  0.975 24.2255 
Mean            25.0065 
Stdev           0.246185 
Quantile  0.025 24.482 
Quantile  0.25  24.8837 
Quantile  0.5   25.0025 
Quantile  0.75  25.1233 
Quantile  0.975 25.5555 
Mean            21.7935 
Stdev           0.214375 
Quantile  0.025 21.318 
Quantile  0.25  21.6897 
Quantile  0.5   21.795 
Quantile  0.75  21.8987 
Quantile  0.975 22.2529 
Mean            20.5965 
Stdev           0.202088 
Quantile  0.025 20.1525 
Quantile  0.25  20.4978 
Quantile  0.5   20.5968 
Quantile  0.75  20.6951 
Quantile  0.975 21.0337 
Mean            21.2132 
Stdev           0.209139 
Quantile  0.025 20.7796 
Quantile  0.25  21.1068 
Quantile  0.5   21.2066 
Quantile  0.75  21.3102 
Quantile  0.975 21.6907 
Mean            19.1072 
Stdev           0.188351 
Quantile  0.025 18.7092 
Quantile  0.25  19.0126 
Quantile  0.5   19.1032 
Quantile  0.75  19.1958 
Quantile  0.975 19.5304 
Mean            20.5989 
Stdev           0.202468 
Quantile  0.025 20.1575 
Quantile  0.25  20.4995 
Quantile  0.5   20.5982 
Quantile  0.75  20.6969 
Quantile  0.975 21.0405 
Mean            15.1848 
Stdev           0.150076 
Quantile  0.025 14.8364 
Quantile  0.25  15.1151 
Quantile  0.5   15.1902 
Quantile  0.75  15.2614 
Quantile  0.975 15.4898 
Mean            7.00998 
Stdev           0.0700733 
Quantile  0.025 6.87361 
Quantile  0.25  6.973 
Quantile  0.5   7.00547 
Quantile  0.75  7.0407 
Quantile  0.975 7.17792 
Mean            8.11337 
Stdev           0.0813952 
Quantile  0.025 7.95784 
Quantile  0.25  8.06997 
Quantile  0.5   8.1074 
Quantile  0.75  8.14852 
Quantile  0.975 8.31079 
Mean            13.6141 
Stdev           0.134906 
Quantile  0.025 13.3432 
Quantile  0.25  13.544 
Quantile  0.5   13.6075 
Quantile  0.75  13.6749 
Quantile  0.975 13.9301 
Mean            20.091 
Stdev           0.197477 
Quantile  0.025 19.6487 
Quantile  0.25  19.9962 
Quantile  0.5   20.0936 
Quantile  0.75  20.1888 
Quantile  0.975 20.5096 
Mean            21.7901 
Stdev           0.214445 
Quantile  0.025 21.3095 
Quantile  0.25  21.6873 
Quantile  0.5   21.793 
Quantile  0.75  21.8963 
Quantile  0.975 22.2445 
Mean            24.494 
Stdev           0.240914 
Quantile  0.025 23.9617 
Quantile  0.25  24.3771 
Quantile  0.5   24.4952 
Quantile  0.75  24.6119 
Quantile  0.975 25.0124 
Mean            23.0837 
Stdev           0.227101 
Quantile  0.025 22.5663 
Quantile  0.25  22.9763 
Quantile  0.5   23.0892 
Quantile  0.75  23.1979 
Quantile  0.975 23.5557 
Mean            19.6949 
Stdev           0.193596 
Quantile  0.025 19.2669 
Quantile  0.25  19.601 
Quantile  0.5   19.696 
Quantile  0.75  19.7898 
Quantile  0.975 20.1111 
Mean            18.3 
Stdev           0.179996 
Quantile  0.025 17.909 
Quantile  0.25  18.2114 
Quantile  0.5   18.299 
Quantile  0.75  18.3867 
Quantile  0.975 18.6941 
Mean            21.1967 
Stdev           0.207975 
Quantile  0.025 20.7402 
Quantile  0.25  21.0951 
Quantile  0.5   21.1969 
Quantile  0.75  21.2981 
Quantile  0.975 21.647 
Mean            17.5019 
Stdev           0.172085 
Quantile  0.025 17.1311 
Quantile  0.25  17.4167 
Quantile  0.5   17.5002 
Quantile  0.75  17.5843 
Quantile  0.975 17.8815 
Mean            16.8077 
Stdev           0.165895 
Quantile  0.025 16.4594 
Quantile  0.25  16.7241 
Quantile  0.5   16.8036 
Quantile  0.75  16.8853 
Quantile  0.975 17.1827 
Mean            22.3948 
Stdev           0.220264 
Quantile  0.025 21.9086 
Quantile  0.25  22.2878 
Quantile  0.5   22.3957 
Quantile  0.75  22.5025 
Quantile  0.975 22.8692 
Mean            20.5996 
Stdev           0.202744 
Quantile  0.025 20.1587 
Quantile  0.25  20.5 
Quantile  0.5   20.5987 
Quantile  0.75  20.6974 
Quantile  0.975 21.043 
Mean            23.894 
Stdev           0.235019 
Quantile  0.025 23.3746 
Quantile  0.25  23.78 
Quantile  0.5   23.8952 
Quantile  0.75  24.009 
Quantile  0.975 24.3996 
Mean            21.9984 
Stdev           0.216497 
Quantile  0.025 21.5257 
Quantile  0.25  21.8923 
Quantile  0.5   21.9979 
Quantile  0.75  22.1032 
Quantile  0.975 22.47 
Mean            11.9113 
Stdev           0.118183 
Quantile  0.025 11.6723 
Quantile  0.25  11.8503 
Quantile  0.5   11.906 
Quantile  0.75  11.9648 
Quantile  0.975 12.1868 
Mean            23.9864 
Stdev           0.236501 
Quantile  0.025 23.4524 
Quantile  0.25  23.8739 
Quantile  0.5   23.9907 
Quantile  0.75  24.1042 
Quantile  0.975 24.4835 
Mean            21.6058 
Stdev           0.212328 
Quantile  0.025 21.1538 
Quantile  0.25  21.4997 
Quantile  0.5   21.6023 
Quantile  0.75  21.7066 
Quantile  0.975 22.0795 
Mean            34.6834 
Stdev           0.341592 
Quantile  0.025 33.9165 
Quantile  0.25  34.5199 
Quantile  0.5   34.6884 
Quantile  0.75  34.8527 
Quantile  0.975 35.4059 
Mean            33.393 
Stdev           0.328641 
Quantile  0.025 32.6688 
Quantile  0.25  33.2333 
Quantile  0.5   33.3941 
Quantile  0.75  33.5534 
Quantile  0.975 34.1021 
Mean            36.1867 
Stdev           0.35625 
Quantile  0.025 35.393 
Quantile  0.25  36.0151 
Quantile  0.5   36.1903 
Quantile  0.75  36.3622 
Quantile  0.975 36.9466 
Mean            28.69 
Stdev           0.282619 
Quantile  0.025 28.0611 
Quantile  0.25  28.5538 
Quantile  0.5   28.6926 
Quantile  0.75  28.829 
Quantile  0.975 29.2937 
Mean            22.9018 
Stdev           0.225026 
Quantile  0.025 22.416 
Quantile  0.25  22.7905 
Quantile  0.5   22.8998 
Quantile  0.75  23.0099 
Quantile  0.975 23.3972 
Mean            27.0795 
Stdev           0.267164 
Quantile  0.025 26.4688 
Quantile  0.25  26.9538 
Quantile  0.5   27.0865 
Quantile  0.75  27.214 
Quantile  0.975 27.633 
Mean            16.5119 
Stdev           0.163053 
Quantile  0.025 16.1763 
Quantile  0.25  16.4285 
Quantile  0.5   16.5061 
Quantile  0.75  16.587 
Quantile  0.975 16.8865 
Mean            18.8997 
Stdev           0.185894 
Quantile  0.025 18.4957 
Quantile  0.25  18.8083 
Quantile  0.5   18.8988 
Quantile  0.75  18.9894 
Quantile  0.975 19.3065 
Mean            15.022 
Stdev           0.150088 
Quantile  0.025 14.7308 
Quantile  0.25  14.9426 
Quantile  0.5   15.0121 
Quantile  0.75  15.0876 
Quantile  0.975 15.3823 
Mean            18.8995 
Stdev           0.185659 
Quantile  0.025 18.4956 
Quantile  0.25  18.8082 
Quantile  0.5   18.8987 
Quantile  0.75  18.9892 
Quantile  0.975 19.3052 
Mean            21.6837 
Stdev           0.213172 
Quantile  0.025 21.1967 
Quantile  0.25  21.583 
Quantile  0.5   21.6892 
Quantile  0.75  21.7912 
Quantile  0.975 22.1252 
Mean            20.3935 
Stdev           0.200366 
Quantile  0.025 19.9485 
Quantile  0.25  20.2966 
Quantile  0.5   20.3951 
Quantile  0.75  20.492 
Quantile  0.975 20.8222 
Mean            18.2015 
Stdev           0.179062 
Quantile  0.025 17.8149 
Quantile  0.25  18.113 
Quantile  0.5   18.1999 
Quantile  0.75  18.2874 
Quantile  0.975 18.5959 
Mean            19.8984 
Stdev           0.195347 
Quantile  0.025 19.4718 
Quantile  0.25  19.8026 
Quantile  0.5   19.898 
Quantile  0.75  19.9932 
Quantile  0.975 20.3236 
Mean            23.0947 
Stdev           0.227004 
Quantile  0.025 22.5938 
Quantile  0.25  22.9844 
Quantile  0.5   23.0957 
Quantile  0.75  23.2057 
Quantile  0.975 23.5837 
Mean            17.5008 
Stdev           0.17194 
Quantile  0.025 17.1286 
Quantile  0.25  17.4159 
Quantile  0.5   17.4995 
Quantile  0.75  17.5835 
Quantile  0.975 17.8784 
Mean            20.1954 
Stdev           0.198759 
Quantile  0.025 19.7568 
Quantile  0.25  20.0989 
Quantile  0.5   20.1962 
Quantile  0.75  20.2925 
Quantile  0.975 20.6237 
Mean            18.2026 
Stdev           0.178784 
Quantile  0.025 17.8184 
Quantile  0.25  18.1138 
Quantile  0.5   18.2006 
Quantile  0.75  18.2881 
Quantile  0.975 18.598 
Mean            13.605 
Stdev           0.133939 
Quantile  0.025 13.322 
Quantile  0.25  13.5378 
Quantile  0.5   13.6022 
Quantile  0.75  13.6682 
Quantile  0.975 13.9059 
Mean            19.5924 
Stdev           0.192892 
Quantile  0.025 19.1622 
Quantile  0.25  19.4996 
Quantile  0.5   19.5945 
Quantile  0.75  19.6876 
Quantile  0.975 20.0033 
Mean            15.205 
Stdev           0.14965 
Quantile  0.025 14.8877 
Quantile  0.25  15.13 
Quantile  0.5   15.2021 
Quantile  0.75  15.2757 
Quantile  0.975 15.5402 
Mean            14.5003 
Stdev           0.142631 
Quantile  0.025 14.1912 
Quantile  0.25  14.4301 
Quantile  0.5   14.4994 
Quantile  0.75  14.569 
Quantile  0.975 14.8132 
Mean            15.6009 
Stdev           0.153469 
Quantile  0.025 15.2691 
Quantile  0.25  15.5252 
Quantile  0.5   15.5997 
Quantile  0.75  15.6747 
Quantile  0.975 15.9384 
Mean            13.9045 
Stdev           0.136847 
Quantile  0.025 13.6143 
Quantile  0.25  13.8359 
Quantile  0.5   13.9019 
Quantile  0.75  13.9692 
Quantile  0.975 14.2109 
Mean            16.6008 
Stdev           0.163408 
Quantile  0.025 16.2473 
Quantile  0.25  16.5202 
Quantile  0.5   16.5996 
Quantile  0.75  16.6793 
Quantile  0.975 16.96 
Mean            14.8078 
Stdev           0.145952 
Quantile  0.025 14.503 
Quantile  0.25  14.7339 
Quantile  0.5   14.8038 
Quantile  0.75  14.8759 
Quantile  0.975 15.1391 
Mean            18.4083 
Stdev           0.181557 
Quantile  0.025 18.0268 
Quantile  0.25  18.3168 
Quantile  0.5   18.4039 
Quantile  0.75  18.4933 
Quantile  0.975 18.8183 
Mean            20.9937 
Stdev           0.206508 
Quantile  0.025 20.5357 
Quantile  0.25  20.8938 
Quantile  0.5   20.9952 
Quantile  0.75  21.095 
Quantile  0.975 21.4362 
Mean            12.7065 
Stdev           0.12515 
Quantile  0.025 12.4449 
Quantile  0.25  12.6432 
Quantile  0.5   12.7032 
Quantile  0.75  12.765 
Quantile  0.975 12.9903 
Mean            14.5052 
Stdev           0.142975 
Quantile  0.025 14.2029 
Quantile  0.25  14.4335 
Quantile  0.5   14.5023 
Quantile  0.75  14.5726 
Quantile  0.975 14.8263 
Mean            13.2072 
Stdev           0.130119 
Quantile  0.025 12.9359 
Quantile  0.25  13.1413 
Quantile  0.5   13.2035 
Quantile  0.75  13.2679 
Quantile  0.975 13.5029 
Mean            13.101 
Stdev           0.128968 
Quantile  0.025 12.8226 
Quantile  0.25  13.0373 
Quantile  0.5   13.0999 
Quantile  0.75  13.1629 
Quantile  0.975 13.3851 
Mean            13.5094 
Stdev           0.133448 
Quantile  0.025 13.2342 
Quantile  0.25  13.4413 
Quantile  0.5   13.5048 
Quantile  0.75  13.5709 
Quantile  0.975 13.8155 
Mean            18.9043 
Stdev           0.185634 
Quantile  0.025 18.5078 
Quantile  0.25  18.8116 
Quantile  0.5   18.9015 
Quantile  0.75  18.9927 
Quantile  0.975 19.3171 
Mean            20.0046 
Stdev           0.196783 
Quantile  0.025 19.5843 
Quantile  0.25  19.9065 
Quantile  0.5   20.0016 
Quantile  0.75  20.0981 
Quantile  0.975 20.4423 
Mean            21.003 
Stdev           0.206408 
Quantile  0.025 20.5594 
Quantile  0.25  20.9005 
Quantile  0.5   21.0006 
Quantile  0.75  21.1017 
Quantile  0.975 21.4594 
Mean            24.6881 
Stdev           0.242697 
Quantile  0.025 24.1433 
Quantile  0.25  24.5718 
Quantile  0.5   24.6917 
Quantile  0.75  24.8086 
Quantile  0.975 25.2012 
Mean            30.7763 
Stdev           0.302606 
Quantile  0.025 30.0843 
Quantile  0.25  30.6336 
Quantile  0.5   30.7844 
Quantile  0.75  30.9291 
Quantile  0.975 31.4023 
Mean            34.8722 
Stdev           0.343339 
Quantile  0.025 34.0855 
Quantile  0.25  34.7107 
Quantile  0.5   34.8818 
Quantile  0.75  35.0457 
Quantile  0.975 35.581 
Mean            26.6069 
Stdev           0.261935 
Quantile  0.025 26.0487 
Quantile  0.25  26.4762 
Quantile  0.5   26.6026 
Quantile  0.75  26.7312 
Quantile  0.975 27.1909 
Mean            25.3011 
Stdev           0.249045 
Quantile  0.025 24.762 
Quantile  0.25  25.1783 
Quantile  0.5   25.2993 
Quantile  0.75  25.4208 
Quantile  0.975 25.8482 
Mean            24.6957 
Stdev           0.242582 
Quantile  0.025 24.1625 
Quantile  0.25  24.5774 
Quantile  0.5   24.6961 
Quantile  0.75  24.814 
Quantile  0.975 25.2203 
Mean            21.1996 
Stdev           0.208019 
Quantile  0.025 20.7474 
Quantile  0.25  21.0972 
Quantile  0.5   21.1986 
Quantile  0.75  21.3002 
Quantile  0.975 21.6544 
Mean            19.3047 
Stdev           0.190033 
Quantile  0.025 18.8993 
Quantile  0.25  19.21 
Quantile  0.5   19.3017 
Quantile  0.75  19.395 
Quantile  0.975 19.728 
Mean            19.9976 
Stdev           0.19655 
Quantile  0.025 19.5672 
Quantile  0.25  19.9015 
Quantile  0.5   19.9975 
Quantile  0.75  20.0931 
Quantile  0.975 20.4243 
Mean            16.6095 
Stdev           0.163984 
Quantile  0.025 16.2682 
Quantile  0.25  16.5264 
Quantile  0.5   16.6047 
Quantile  0.75  16.6857 
Quantile  0.975 16.9829 
Mean            14.4082 
Stdev           0.142154 
Quantile  0.025 14.1123 
Quantile  0.25  14.3361 
Quantile  0.5   14.404 
Quantile  0.75  14.4743 
Quantile  0.975 14.7318 
Mean            19.395 
Stdev           0.190889 
Quantile  0.025 18.973 
Quantile  0.25  19.3025 
Quantile  0.5   19.396 
Quantile  0.75  19.4885 
Quantile  0.975 19.8055 
Mean            19.7062 
Stdev           0.193708 
Quantile  0.025 19.2953 
Quantile  0.25  19.6092 
Quantile  0.5   19.7026 
Quantile  0.75  19.7979 
Quantile  0.975 20.1397 
Mean            20.5072 
Stdev           0.201987 
Quantile  0.025 20.0796 
Quantile  0.25  20.4059 
Quantile  0.5   20.5031 
Quantile  0.75  20.6024 
Quantile  0.975 20.9603 
Mean            25.0028 
Stdev           0.245691 
Quantile  0.025 24.4736 
Quantile  0.25  24.881 
Quantile  0.5   25.0003 
Quantile  0.75  25.1205 
Quantile  0.975 25.5449 
Mean            23.3967 
Stdev           0.229823 
Quantile  0.025 22.8928 
Quantile  0.25  23.2844 
Quantile  0.5   23.3968 
Quantile  0.75  23.5086 
Quantile  0.975 23.895 
Mean            18.9127 
Stdev           0.186785 
Quantile  0.025 18.5269 
Quantile  0.25  18.8175 
Quantile  0.5   18.9065 
Quantile  0.75  18.999 
Quantile  0.975 19.3407 
Mean            35.3872 
Stdev           0.34815 
Quantile  0.025 34.6119 
Quantile  0.25  35.2194 
Quantile  0.5   35.3906 
Quantile  0.75  35.5587 
Quantile  0.975 36.1301 
Mean            24.7069 
Stdev           0.243577 
Quantile  0.025 24.1885 
Quantile  0.25  24.5853 
Quantile  0.5   24.7027 
Quantile  0.75  24.8222 
Quantile  0.975 25.2507 
Mean            31.5866 
Stdev           0.310439 
Quantile  0.025 30.8925 
Quantile  0.25  31.4374 
Quantile  0.5   31.5905 
Quantile  0.75  31.7402 
Quantile  0.975 32.2459 
Mean            23.2945 
Stdev           0.228832 
Quantile  0.025 22.7893 
Quantile  0.25  23.1833 
Quantile  0.5   23.2955 
Quantile  0.75  23.4065 
Quantile  0.975 23.7871 
Mean            19.6004 
Stdev           0.192675 
Quantile  0.025 19.1826 
Quantile  0.25  19.5054 
Quantile  0.5   19.5992 
Quantile  0.75  19.6932 
Quantile  0.975 20.0229 
Mean            18.6996 
Stdev           0.184043 
Quantile  0.025 18.2994 
Quantile  0.25  18.6092 
Quantile  0.5   18.6988 
Quantile  0.75  18.7884 
Quantile  0.975 19.1021 
Mean            16.009 
Stdev           0.157834 
Quantile  0.025 15.6802 
Quantile  0.25  15.9289 
Quantile  0.5   16.0044 
Quantile  0.75  16.0824 
Quantile  0.975 16.368 
Mean            22.2005 
Stdev           0.218373 
Quantile  0.025 21.7271 
Quantile  0.25  22.0929 
Quantile  0.5   22.1991 
Quantile  0.75  22.3056 
Quantile  0.975 22.6795 
Mean            25.0083 
Stdev           0.246458 
Quantile  0.025 24.486 
Quantile  0.25  24.8849 
Quantile  0.5   25.0036 
Quantile  0.75  25.1247 
Quantile  0.975 25.5606 
Mean            32.9961 
Stdev           0.324942 
Quantile  0.025 32.2844 
Quantile  0.25  32.8374 
Quantile  0.5   32.9959 
Quantile  0.75  33.1537 
Quantile  0.975 33.7018 
Mean            23.5025 
Stdev           0.231229 
Quantile  0.025 23.0043 
Quantile  0.25  23.3881 
Quantile  0.5   23.5002 
Quantile  0.75  23.6133 
Quantile  0.975 24.0127 
Mean            19.4052 
Stdev           0.191172 
Quantile  0.025 18.9982 
Quantile  0.25  19.3098 
Quantile  0.5   19.402 
Quantile  0.75  19.4958 
Quantile  0.975 19.8318 
Mean            21.9948 
Stdev           0.216065 
Quantile  0.025 21.5179 
Quantile  0.25  21.8898 
Quantile  0.5   21.9958 
Quantile  0.75  22.1006 
Quantile  0.975 22.46 
Mean            17.404 
Stdev           0.171659 
Quantile  0.025 17.0373 
Quantile  0.25  17.3186 
Quantile  0.5   17.4014 
Quantile  0.75  17.4855 
Quantile  0.975 17.786 
Mean            20.9003 
Stdev           0.205581 
Quantile  0.025 20.4544 
Quantile  0.25  20.7991 
Quantile  0.5   20.8991 
Quantile  0.75  20.9993 
Quantile  0.975 21.351 
Mean            24.2018 
Stdev           0.238403 
Quantile  0.025 23.6868 
Quantile  0.25  24.0841 
Quantile  0.5   24.1998 
Quantile  0.75  24.3161 
Quantile  0.975 24.7267 
Mean            21.6987 
Stdev           0.213026 
Quantile  0.025 21.2343 
Quantile  0.25  21.5941 
Quantile  0.5   21.6981 
Quantile  0.75  21.8019 
Quantile  0.975 22.1632 
Mean            22.802 
Stdev           0.224469 
Quantile  0.025 22.3175 
Quantile  0.25  22.691 
Quantile  0.5   22.7999 
Quantile  0.75  22.9096 
Quantile  0.975 23.2965 
Mean            23.4104 
Stdev           0.230587 
Quantile  0.025 22.9258 
Quantile  0.25  23.2942 
Quantile  0.5   23.4048 
Quantile  0.75  23.5185 
Quantile  0.975 23.9309 
Mean            24.1052 
Stdev           0.237109 
Quantile  0.025 23.5983 
Quantile  0.25  23.9871 
Quantile  0.5   24.1018 
Quantile  0.75  24.218 
Quantile  0.975 24.6322 
Mean            21.4053 
Stdev           0.210716 
Quantile  0.025 20.956 
Quantile  0.25  21.3002 
Quantile  0.5   21.402 
Quantile  0.75  21.5054 
Quantile  0.975 21.8748 
Mean            20.0078 
Stdev           0.197121 
Quantile  0.025 19.5918 
Quantile  0.25  19.9088 
Quantile  0.5   20.0035 
Quantile  0.75  20.1005 
Quantile  0.975 20.4512 
Mean            20.8046 
Stdev           0.204647 
Quantile  0.025 20.3673 
Quantile  0.25  20.7026 
Quantile  0.5   20.8016 
Quantile  0.75  20.902 
Quantile  0.975 21.2596 
Mean            21.2063 
Stdev           0.20855 
Quantile  0.025 20.7631 
Quantile  0.25  21.1019 
Quantile  0.5   21.2025 
Quantile  0.75  21.3051 
Quantile  0.975 21.6723 
Mean            20.3035 
Stdev           0.199801 
Quantile  0.025 19.875 
Quantile  0.25  20.2042 
Quantile  0.5   20.301 
Quantile  0.75  20.3988 
Quantile  0.975 20.7463 
Mean            27.9935 
Stdev           0.274993 
Quantile  0.025 27.3865 
Quantile  0.25  27.8598 
Quantile  0.5   27.9947 
Quantile  0.75  28.128 
Quantile  0.975 28.5856 
Mean            23.9077 
Stdev           0.235599 
Quantile  0.025 23.4081 
Quantile  0.25  23.7898 
Quantile  0.5   23.9033 
Quantile  0.75  24.019 
Quantile  0.975 24.4354 
Mean            24.7962 
Stdev           0.243867 
Quantile  0.025 24.2608 
Quantile  0.25  24.6772 
Quantile  0.5   24.7964 
Quantile  0.75  24.9148 
Quantile  0.975 25.3244 
Mean            22.904 
Stdev           0.224862 
Quantile  0.025 22.422 
Quantile  0.25  22.7921 
Quantile  0.5   22.9011 
Quantile  0.75  23.0114 
Quantile  0.975 23.4023 
Mean            23.9084 
Stdev           0.235488 
Quantile  0.025 23.4099 
Quantile  0.25  23.7903 
Quantile  0.5   23.9036 
Quantile  0.75  24.0194 
Quantile  0.975 24.4366 
Mean            26.5988 
Stdev           0.261287 
Quantile  0.025 26.0296 
Quantile  0.25  26.4704 
Quantile  0.5   26.5978 
Quantile  0.75  26.7252 
Quantile  0.975 27.169 
Mean            22.5056 
Stdev           0.221409 
Quantile  0.025 22.0334 
Quantile  0.25  22.3951 
Quantile  0.5   22.5021 
Quantile  0.75  22.6108 
Quantile  0.975 22.9988 
Mean            22.2014 
Stdev           0.218266 
Quantile  0.025 21.7297 
Quantile  0.25  22.0936 
Quantile  0.5   22.1996 
Quantile  0.75  22.3063 
Quantile  0.975 22.6815 
Mean            23.6097 
Stdev           0.232796 
Quantile  0.025 23.1193 
Quantile  0.25  23.4927 
Quantile  0.5   23.6045 
Quantile  0.75  23.7191 
Quantile  0.975 24.1342 
Mean            28.6979 
Stdev           0.282247 
Quantile  0.025 28.0817 
Quantile  0.25  28.5596 
Quantile  0.5   28.6972 
Quantile  0.75  28.8346 
Quantile  0.975 29.3127 
Mean            22.607 
Stdev           0.222213 
Quantile  0.025 22.1352 
Quantile  0.25  22.4957 
Quantile  0.5   22.6029 
Quantile  0.75  22.7122 
Quantile  0.975 23.1039 
Mean            22.0005 
Stdev           0.216016 
Quantile  0.025 21.5324 
Quantile  0.25  21.894 
Quantile  0.5   21.9991 
Quantile  0.75  22.1047 
Quantile  0.975 22.4743 
Mean            22.9029 
Stdev           0.224934 
Quantile  0.025 22.4189 
Quantile  0.25  22.7913 
Quantile  0.5   22.9005 
Quantile  0.75  23.0106 
Quantile  0.975 23.3996 
Mean            24.9889 
Stdev           0.245615 
Quantile  0.025 24.4389 
Quantile  0.25  24.8709 
Quantile  0.5   24.9921 
Quantile  0.75  25.1106 
Quantile  0.975 25.5096 
Mean            20.6115 
Stdev           0.203212 
Quantile  0.025 20.1882 
Quantile  0.25  20.5085 
Quantile  0.5   20.6057 
Quantile  0.75  20.7062 
Quantile  0.975 21.0737 
Mean            28.3903 
Stdev           0.279472 
Quantile  0.025 27.7688 
Quantile  0.25  28.2555 
Quantile  0.5   28.3928 
Quantile  0.75  28.5278 
Quantile  0.975 28.9876 
Mean            21.4045 
Stdev           0.210673 
Quantile  0.025 20.954 
Quantile  0.25  21.2997 
Quantile  0.5   21.4015 
Quantile  0.75  21.5048 
Quantile  0.975 21.8727 
Mean            38.6968 
Stdev           0.380366 
Quantile  0.025 37.866 
Quantile  0.25  38.5104 
Quantile  0.5   38.696 
Quantile  0.75  38.8812 
Quantile  0.975 39.5248 
Mean            43.7756 
Stdev           0.430515 
Quantile  0.025 42.8044 
Quantile  0.25  43.5703 
Quantile  0.5   43.7833 
Quantile  0.75  43.9903 
Quantile  0.975 44.6808 
Mean            33.1978 
Stdev           0.326106 
Quantile  0.025 32.4864 
Quantile  0.25  33.0378 
Quantile  0.5   33.1969 
Quantile  0.75  33.3558 
Quantile  0.975 33.9085 
Mean            27.4756 
Stdev           0.27017 
Quantile  0.025 26.8533 
Quantile  0.25  27.349 
Quantile  0.5   27.4841 
Quantile  0.75  27.613 
Quantile  0.975 28.0294 
Mean            26.5018 
Stdev           0.260237 
Quantile  0.025 25.9396 
Quantile  0.25  26.3731 
Quantile  0.5   26.4997 
Quantile  0.75  26.6269 
Quantile  0.975 27.0743 
Mean            18.6158 
Stdev           0.184229 
Quantile  0.025 18.2406 
Quantile  0.25  18.5211 
Quantile  0.5   18.6083 
Quantile  0.75  18.6999 
Quantile  0.975 19.0427 
Mean            19.3061 
Stdev           0.189997 
Quantile  0.025 18.9029 
Quantile  0.25  19.2109 
Quantile  0.5   19.3025 
Quantile  0.75  19.3959 
Quantile  0.975 19.7312 
Mean            20.1045 
Stdev           0.197643 
Quantile  0.025 19.6823 
Quantile  0.25  20.006 
Quantile  0.5   20.1016 
Quantile  0.75  20.1986 
Quantile  0.975 20.544 
Mean            19.4909 
Stdev           0.191709 
Quantile  0.025 19.0609 
Quantile  0.25  19.399 
Quantile  0.5   19.4936 
Quantile  0.75  19.5859 
Quantile  0.975 19.8967 
Mean            19.4987 
Stdev           0.191772 
Quantile  0.025 19.0802 
Quantile  0.25  19.4047 
Quantile  0.5   19.4982 
Quantile  0.75  19.5915 
Quantile  0.975 19.9166 
Mean            20.3957 
Stdev           0.200349 
Quantile  0.025 19.9541 
Quantile  0.25  20.2982 
Quantile  0.5   20.3964 
Quantile  0.75  20.4936 
Quantile  0.975 20.8278 
Mean            19.8091 
Stdev           0.19514 
Quantile  0.025 19.3995 
Quantile  0.25  19.7107 
Quantile  0.5   19.8043 
Quantile  0.75  19.9006 
Quantile  0.975 20.2501 
Mean            19.4045 
Stdev           0.19126 
Quantile  0.025 18.9962 
Quantile  0.25  19.3093 
Quantile  0.5   19.4016 
Quantile  0.75  19.4953 
Quantile  0.975 19.8303 
Mean            21.6963 
Stdev           0.213381 
Quantile  0.025 21.2273 
Quantile  0.25  21.5923 
Quantile  0.5   21.6967 
Quantile  0.75  21.8002 
Quantile  0.975 22.1579 
Mean            22.8061 
Stdev           0.224383 
Quantile  0.025 22.3282 
Quantile  0.25  22.694 
Quantile  0.5   22.8023 
Quantile  0.75  22.9125 
Quantile  0.975 23.3065 
Mean            18.799 
Stdev           0.184894 
Quantile  0.025 18.3961 
Quantile  0.25  18.7083 
Quantile  0.5   18.7984 
Quantile  0.75  18.8884 
Quantile  0.975 19.2024 
Mean            18.704 
Stdev           0.183977 
Quantile  0.025 18.3106 
Quantile  0.25  18.6124 
Quantile  0.5   18.7013 
Quantile  0.75  18.7915 
Quantile  0.975 19.1128 
Mean            18.5094 
Stdev           0.182192 
Quantile  0.025 18.1285 
Quantile  0.25  18.4172 
Quantile  0.5   18.5045 
Quantile  0.75  18.5946 
Quantile  0.975 18.9224 
Mean            18.3036 
Stdev           0.180027 
Quantile  0.025 17.9183 
Quantile  0.25  18.214 
Quantile  0.5   18.3011 
Quantile  0.75  18.3894 
Quantile  0.975 18.7033 
Mean            21.1978 
Stdev           0.208478 
Quantile  0.025 20.7418 
Quantile  0.25  21.0958 
Quantile  0.5   21.1976 
Quantile  0.75  21.299 
Quantile  0.975 21.6511 
Mean            19.2048 
Stdev           0.188939 
Quantile  0.025 18.802 
Quantile  0.25  19.1106 
Quantile  0.5   19.2018 
Quantile  0.75  19.2946 
Quantile  0.975 19.6259 
Mean            20.3984 
Stdev           0.200618 
Quantile  0.025 19.9604 
Quantile  0.25  20.3001 
Quantile  0.5   20.398 
Quantile  0.75  20.4956 
Quantile  0.975 20.8353 
Mean            19.3034 
Stdev           0.189844 
Quantile  0.025 18.8964 
Quantile  0.25  19.209 
Quantile  0.5   19.301 
Quantile  0.75  19.394 
Quantile  0.975 19.7242 
Mean            21.9949 
Stdev           0.215819 
Quantile  0.025 21.5186 
Quantile  0.25  21.8899 
Quantile  0.5   21.9958 
Quantile  0.75  22.1006 
Quantile  0.975 22.4595 
Mean            20.2983 
Stdev           0.199388 
Quantile  0.025 19.8628 
Quantile  0.25  20.2006 
Quantile  0.5   20.2979 
Quantile  0.75  20.395 
Quantile  0.975 20.7323 
Mean            20.4966 
Stdev           0.201849 
Quantile  0.025 20.0532 
Quantile  0.25  20.3983 
Quantile  0.5   20.4969 
Quantile  0.75  20.5948 
Quantile  0.975 20.9336 
Mean            17.3048 
Stdev           0.170267 
Quantile  0.025 16.9425 
Quantile  0.25  17.2197 
Quantile  0.5   17.3019 
Quantile  0.75  17.3855 
Quantile  0.975 17.6848 
Mean            18.802 
Stdev           0.184984 
Quantile  0.025 18.4034 
Quantile  0.25  18.7105 
Quantile  0.5   18.8002 
Quantile  0.75  18.8906 
Quantile  0.975 19.2102 
Mean            21.3869 
Stdev           0.210649 
Quantile  0.025 20.9099 
Quantile  0.25  21.2868 
Quantile  0.5   21.3912 
Quantile  0.75  21.4922 
Quantile  0.975 21.828 
Mean            15.6977 
Stdev           0.154106 
Quantile  0.025 15.3596 
Quantile  0.25  15.6224 
Quantile  0.5   15.6978 
Quantile  0.75  15.7727 
Quantile  0.975 16.0315 
Mean            16.1958 
Stdev           0.159024 
Quantile  0.025 15.8442 
Quantile  0.25  16.1186 
Quantile  0.5   16.1967 
Quantile  0.75  16.2738 
Quantile  0.975 16.5376 
Mean            18.0023 
Stdev           0.177018 
Quantile  0.025 17.6215 
Quantile  0.25  17.9146 
Quantile  0.5   18.0004 
Quantile  0.75  18.087 
Quantile  0.975 18.3934 
Mean            14.3048 
Stdev           0.140885 
Quantile  0.025 14.0063 
Quantile  0.25  14.2342 
Quantile  0.5   14.302 
Quantile  0.75  14.3713 
Quantile  0.975 14.6205 
Mean            19.207 
Stdev           0.188966 
Quantile  0.025 18.8075 
Quantile  0.25  19.1121 
Quantile  0.5   19.2031 
Quantile  0.75  19.2961 
Quantile  0.975 19.6312 
Mean            19.6002 
Stdev           0.192915 
Quantile  0.025 19.1816 
Quantile  0.25  19.5053 
Quantile  0.5   19.5991 
Quantile  0.75  19.6931 
Quantile  0.975 20.023 
Mean            22.9894 
Stdev           0.226114 
Quantile  0.025 22.4824 
Quantile  0.25  22.8809 
Quantile  0.5   22.9925 
Quantile  0.75  23.1014 
Quantile  0.975 23.4682 
Mean            18.4022 
Stdev           0.180836 
Quantile  0.025 18.013 
Quantile  0.25  18.3126 
Quantile  0.5   18.4003 
Quantile  0.75  18.4888 
Quantile  0.975 18.8015 
Mean            15.6071 
Stdev           0.153835 
Quantile  0.025 15.284 
Quantile  0.25  15.5295 
Quantile  0.5   15.6033 
Quantile  0.75  15.6792 
Quantile  0.975 15.9546 
Mean            18.1088 
Stdev           0.178427 
Quantile  0.025 17.735 
Quantile  0.25  18.0187 
Quantile  0.5   18.1042 
Quantile  0.75  18.1923 
Quantile  0.975 18.5127 
Mean            17.4096 
Stdev           0.171427 
Quantile  0.025 17.0522 
Quantile  0.25  17.3226 
Quantile  0.5   17.4047 
Quantile  0.75  17.4895 
Quantile  0.975 17.7992 
Mean            17.1047 
Stdev           0.168407 
Quantile  0.025 16.7464 
Quantile  0.25  17.0206 
Quantile  0.5   17.1019 
Quantile  0.75  17.1845 
Quantile  0.975 17.4807 
Mean            13.3259 
Stdev           0.134518 
Quantile  0.025 13.0749 
Quantile  0.25  13.2533 
Quantile  0.5   13.3144 
Quantile  0.75  13.3828 
Quantile  0.975 13.657 
Mean            17.7954 
Stdev           0.175032 
Quantile  0.025 17.4085 
Quantile  0.25  17.7105 
Quantile  0.5   17.7964 
Quantile  0.75  17.8811 
Quantile  0.975 18.1718 
Mean            14.0097 
Stdev           0.138307 
Quantile  0.025 13.7244 
Quantile  0.25  13.9391 
Quantile  0.5   14.0049 
Quantile  0.75  14.0735 
Quantile  0.975 14.3269 
Mean            14.3936 
Stdev           0.141486 
Quantile  0.025 14.0768 
Quantile  0.25  14.3256 
Quantile  0.5   14.3954 
Quantile  0.75  14.4637 
Quantile  0.975 14.6936 
Mean            13.419 
Stdev           0.133739 
Quantile  0.025 13.1587 
Quantile  0.25  13.3484 
Quantile  0.5   13.4105 
Quantile  0.75  13.4778 
Quantile  0.975 13.7393 
Mean            15.6006 
Stdev           0.153459 
Quantile  0.025 15.2684 
Quantile  0.25  15.525 
Quantile  0.5   15.5995 
Quantile  0.75  15.6744 
Quantile  0.975 15.9376 
Mean            11.8025 
Stdev           0.116243 
Quantile  0.025 11.554 
Quantile  0.25  11.7447 
Quantile  0.5   11.8008 
Quantile  0.75  11.8578 
Quantile  0.975 12.0609 
Mean            13.8152 
Stdev           0.137131 
Quantile  0.025 13.5414 
Quantile  0.25  13.7438 
Quantile  0.5   13.8082 
Quantile  0.75  13.8768 
Quantile  0.975 14.1378 
Mean            15.6038 
Stdev           0.1536 
Quantile  0.025 15.2761 
Quantile  0.25  15.5272 
Quantile  0.5   15.6014 
Quantile  0.75  15.6767 
Quantile  0.975 15.9459 
Mean            14.5943 
Stdev           0.143507 
Quantile  0.025 14.274 
Quantile  0.25  14.5252 
Quantile  0.5   14.5958 
Quantile  0.75  14.6651 
Quantile  0.975 14.8997 
Mean            17.7869 
Stdev           0.175234 
Quantile  0.025 17.3869 
Quantile  0.25  17.7042 
Quantile  0.5   17.7914 
Quantile  0.75  17.8751 
Quantile  0.975 18.1504 
Mean            15.4061 
Stdev           0.151692 
Quantile  0.025 15.0861 
Quantile  0.25  15.3298 
Quantile  0.5   15.4027 
Quantile  0.75  15.4775 
Quantile  0.975 15.7473 
Mean            21.4966 
Stdev           0.211284 
Quantile  0.025 21.0328 
Quantile  0.25  21.3935 
Quantile  0.5   21.4969 
Quantile  0.75  21.5995 
Quantile  0.975 21.9542 
Mean            19.5774 
Stdev           0.193401 
Quantile  0.025 19.1246 
Quantile  0.25  19.4884 
Quantile  0.5   19.5857 
Quantile  0.75  19.6771 
Quantile  0.975 19.9659 
Mean            15.3115 
Stdev           0.150921 
Quantile  0.025 15.0017 
Quantile  0.25  15.2341 
Quantile  0.5   15.3059 
Quantile  0.75  15.381 
Quantile  0.975 15.6588 
Mean            19.3901 
Stdev           0.190875 
Quantile  0.025 18.9608 
Quantile  0.25  19.2989 
Quantile  0.5   19.3931 
Quantile  0.75  19.4849 
Quantile  0.975 19.793 
Mean            17.0065 
Stdev           0.167446 
Quantile  0.025 16.6531 
Quantile  0.25  16.9224 
Quantile  0.5   17.0029 
Quantile  0.75  17.0854 
Quantile  0.975 17.3831 
Mean            15.6128 
Stdev           0.154447 
Quantile  0.025 15.2974 
Quantile  0.25  15.5335 
Quantile  0.5   15.6067 
Quantile  0.75  15.6834 
Quantile  0.975 15.9699 
Mean            13.1091 
Stdev           0.129582 
Quantile  0.025 12.8419 
Quantile  0.25  13.043 
Quantile  0.5   13.1047 
Quantile  0.75  13.1689 
Quantile  0.975 13.4065 
Mean            41.256 
Stdev           0.408044 
Quantile  0.025 40.3053 
Quantile  0.25  41.0675 
Quantile  0.5   41.2719 
Quantile  0.75  41.465 
Quantile  0.975 42.0813 
Mean            24.3024 
Stdev           0.238804 
Quantile  0.025 23.7876 
Quantile  0.25  24.1842 
Quantile  0.5   24.3001 
Quantile  0.75  24.4169 
Quantile  0.975 24.8289 
Mean            23.3044 
Stdev           0.229347 
Quantile  0.025 22.813 
Quantile  0.25  23.1904 
Quantile  0.5   23.3013 
Quantile  0.75  23.4137 
Quantile  0.975 23.8132 
Mean            26.9744 
Stdev           0.266112 
Quantile  0.025 26.359 
Quantile  0.25  26.8505 
Quantile  0.5   26.9835 
Quantile  0.75  27.11 
Quantile  0.975 27.5176 
Mean            49.9801 
Stdev           0.492095 
Quantile  0.025 48.8813 
Quantile  0.25  49.7435 
Quantile  0.5   49.9856 
Quantile  0.75  50.2229 
Quantile  0.975 51.0271 
Mean            49.9493 
Stdev           0.493154 
Quantile  0.025 48.8038 
Quantile  0.25  49.7205 
Quantile  0.5   49.9675 
Quantile  0.75  50.2013 
Quantile  0.975 50.9504 
Mean            49.982 
Stdev           0.491191 
Quantile  0.025 48.8883 
Quantile  0.25  49.745 
Quantile  0.5   49.9867 
Quantile  0.75  50.2241 
Quantile  0.975 51.0299 
Mean            22.7084 
Stdev           0.223561 
Quantile  0.025 22.2359 
Quantile  0.25  22.5961 
Quantile  0.5   22.7037 
Quantile  0.75  22.8138 
Quantile  0.975 23.2105 
Mean            25.0051 
Stdev           0.246264 
Quantile  0.025 24.4781 
Quantile  0.25  24.8826 
Quantile  0.5   25.0017 
Quantile  0.75  25.1223 
Quantile  0.975 25.5521 
Mean            49.9634 
Stdev           0.491678 
Quantile  0.025 48.8413 
Quantile  0.25  49.7312 
Quantile  0.5   49.9758 
Quantile  0.75  50.211 
Quantile  0.975 50.9833 
Mean            23.7982 
Stdev           0.233515 
Quantile  0.025 23.2885 
Quantile  0.25  23.6836 
Quantile  0.5   23.7977 
Quantile  0.75  23.9115 
Quantile  0.975 24.3067 
Mean            23.7971 
Stdev           0.233894 
Quantile  0.025 23.2849 
Quantile  0.25  23.6827 
Quantile  0.5   23.797 
Quantile  0.75  23.9108 
Quantile  0.975 24.3049 
Mean            22.3036 
Stdev           0.219618 
Quantile  0.025 21.8322 
Quantile  0.25  22.1946 
Quantile  0.5   22.3009 
Quantile  0.75  22.4084 
Quantile  0.975 22.7901 
Mean            17.419 
Stdev           0.172873 
Quantile  0.025 17.0735 
Quantile  0.25  17.3291 
Quantile  0.5   17.4102 
Quantile  0.75  17.4967 
Quantile  0.975 17.8255 
Mean            19.1008 
Stdev           0.18755 
Quantile  0.025 18.6948 
Quantile  0.25  19.0082 
Quantile  0.5   19.0994 
Quantile  0.75  19.1911 
Quantile  0.975 19.5126 
Mean            23.09 
Stdev           0.227222 
Quantile  0.025 22.5815 
Quantile  0.25  22.9809 
Quantile  0.5   23.0929 
Quantile  0.75  23.2024 
Quantile  0.975 23.5722 
Mean            23.605 
Stdev           0.232332 
Quantile  0.025 23.1083 
Quantile  0.25  23.4894 
Quantile  0.5   23.6017 
Quantile  0.75  23.7156 
Quantile  0.975 24.1214 
Mean            22.5928 
Stdev           0.222109 
Quantile  0.025 22.0996 
Quantile  0.25  22.4854 
Quantile  0.5   22.5946 
Quantile  0.75  22.702 
Quantile  0.975 23.0681 
Mean            29.386 
Stdev           0.288872 
Quantile  0.025 28.7378 
Quantile  0.25  29.2475 
Quantile  0.5   29.3902 
Quantile  0.75  29.5294 
Quantile  0.975 29.9971 
Mean            23.2133 
Stdev           0.228899 
Quantile  0.025 22.737 
Quantile  0.25  23.0972 
Quantile  0.5   23.2066 
Quantile  0.75  23.3198 
Quantile  0.975 23.7344 
Mean            24.6123 
Stdev           0.242688 
Quantile  0.025 24.1045 
Quantile  0.25  24.4897 
Quantile  0.5   24.6059 
Quantile  0.75  24.7257 
Quantile  0.975 25.1622 
Mean            29.9032 
Stdev           0.294022 
Quantile  0.025 29.2698 
Quantile  0.25  29.7576 
Quantile  0.5   29.9003 
Quantile  0.75  30.0441 
Quantile  0.975 30.5519 
Mean            37.1746 
Stdev           0.365881 
Quantile  0.025 36.3422 
Quantile  0.25  37.0013 
Quantile  0.5   37.183 
Quantile  0.75  37.3583 
Quantile  0.975 37.9365 
Mean            39.8133 
Stdev           0.391428 
Quantile  0.025 38.984 
Quantile  0.25  39.617 
Quantile  0.5   39.8057 
Quantile  0.75  39.9984 
Quantile  0.975 40.6903 
Mean            36.1622 
Stdev           0.357575 
Quantile  0.025 35.3301 
Quantile  0.25  35.9968 
Quantile  0.5   36.1759 
Quantile  0.75  36.3451 
Quantile  0.975 36.8866 
Mean            37.9006 
Stdev           0.373043 
Quantile  0.025 37.0915 
Quantile  0.25  37.717 
Quantile  0.5   37.8983 
Quantile  0.75  38.0802 
Quantile  0.975 38.7185 
Mean            32.4991 
Stdev           0.319448 
Quantile  0.025 31.8042 
Quantile  0.25  32.3421 
Quantile  0.5   32.4978 
Quantile  0.75  32.6535 
Quantile  0.975 33.1973 
Mean            26.3934 
Stdev           0.259767 
Quantile  0.025 25.8194 
Quantile  0.25  26.2675 
Quantile  0.5   26.3947 
Quantile  0.75  26.5205 
Quantile  0.975 26.9523 
Mean            29.5944 
Stdev           0.290705 
Quantile  0.025 28.9546 
Quantile  0.25  29.4527 
Quantile  0.5   29.5951 
Quantile  0.75  29.7363 
Quantile  0.975 30.2223 
Mean            49.9642 
Stdev           0.491903 
Quantile  0.025 48.8428 
Quantile  0.25  49.7318 
Quantile  0.5   49.9762 
Quantile  0.75  50.2116 
Quantile  0.975 50.986 
Mean            32.0052 
Stdev           0.314744 
Quantile  0.025 31.3297 
Quantile  0.25  31.8489 
Quantile  0.5   32.0013 
Quantile  0.75  32.1555 
Quantile  0.975 32.7022 
Mean            29.8013 
Stdev           0.29334 
Quantile  0.025 29.1663 
Quantile  0.25  29.6567 
Quantile  0.5   29.7992 
Quantile  0.75  29.9423 
Quantile  0.975 30.4457 
Mean            34.9065 
Stdev           0.343103 
Quantile  0.025 34.1714 
Quantile  0.25  34.7358 
Quantile  0.5   34.9019 
Quantile  0.75  35.0701 
Quantile  0.975 35.6674 
Mean            36.9771 
Stdev           0.363774 
Quantile  0.025 36.1531 
Quantile  0.25  36.8042 
Quantile  0.5   36.9846 
Quantile  0.75  37.1591 
Quantile  0.975 37.7384 
Mean            30.5066 
Stdev           0.300666 
Quantile  0.025 29.8639 
Quantile  0.25  30.3571 
Quantile  0.5   30.5023 
Quantile  0.75  30.6495 
Quantile  0.975 31.1752 
Mean            36.3951 
Stdev           0.35794 
Quantile  0.025 35.6103 
Quantile  0.25  36.2203 
Quantile  0.5   36.3952 
Quantile  0.75  36.5691 
Quantile  0.975 37.1714 
Mean            31.0975 
Stdev           0.306043 
Quantile  0.025 30.429 
Quantile  0.25  30.9476 
Quantile  0.5   31.0968 
Quantile  0.75  31.2457 
Quantile  0.975 31.7638 
Mean            29.1018 
Stdev           0.287075 
Quantile  0.025 28.481 
Quantile  0.25  28.9603 
Quantile  0.5   29.0995 
Quantile  0.75  29.2394 
Quantile  0.975 29.7334 
Mean            49.9534 
Stdev           0.492198 
Quantile  0.025 48.8162 
Quantile  0.25  49.7237 
Quantile  0.5   49.9699 
Quantile  0.75  50.2041 
Quantile  0.975 50.959 
Mean            33.3122 
Stdev           0.328143 
Quantile  0.025 32.6184 
Quantile  0.25  33.1475 
Quantile  0.5   33.3054 
Quantile  0.75  33.4668 
Quantile  0.975 34.049 
Mean            30.3081 
Stdev           0.298779 
Quantile  0.025 29.6717 
Quantile  0.25  30.1591 
Quantile  0.5   30.3031 
Quantile  0.75  30.4496 
Quantile  0.975 30.9747 
Mean            34.6034 
Stdev           0.340887 
Quantile  0.025 33.8682 
Quantile  0.25  34.4349 
Quantile  0.5   34.6001 
Quantile  0.75  34.7666 
Quantile  0.975 35.3551 
Mean            34.8955 
Stdev           0.343193 
Quantile  0.025 34.1435 
Quantile  0.25  34.7279 
Quantile  0.5   34.8955 
Quantile  0.75  35.0623 
Quantile  0.975 35.6402 
Mean            32.9029 
Stdev           0.323907 
Quantile  0.025 32.204 
Quantile  0.25  32.7428 
Quantile  0.5   32.8999 
Quantile  0.75  33.0582 
Quantile  0.975 33.6167 
Mean            24.099 
Stdev           0.237024 
Quantile  0.025 23.5829 
Quantile  0.25  23.9827 
Quantile  0.5   24.0981 
Quantile  0.75  24.2136 
Quantile  0.975 24.6166 
Mean            42.2814 
Stdev           0.415109 
Quantile  0.025 41.3523 
Quantile  0.25  42.0819 
Quantile  0.5   42.2868 
Quantile  0.75  42.4872 
Quantile  0.975 43.1617 
Mean            48.4829 
Stdev           0.476976 
Quantile  0.025 47.4212 
Quantile  0.25  48.2528 
Quantile  0.5   48.4873 
Quantile  0.75  48.7177 
Quantile  0.975 49.5011 
Mean            49.9513 
Stdev           0.49216 
Quantile  0.025 48.8114 
Quantile  0.25  49.7222 
Quantile  0.5   49.9687 
Quantile  0.75  50.2026 
Quantile  0.975 50.9536 
Mean            22.6109 
Stdev           0.222785 
Quantile  0.025 22.1442 
Quantile  0.25  22.4984 
Quantile  0.5   22.6052 
Quantile  0.75  22.7152 
Quantile  0.975 23.1152 
Mean            24.4172 
Stdev           0.241077 
Quantile  0.025 23.9205 
Quantile  0.25  24.2941 
Quantile  0.5   24.4088 
Quantile  0.75  24.5284 
Quantile  0.975 24.9706 
Mean            22.507 
Stdev           0.221502 
Quantile  0.025 22.0369 
Quantile  0.25  22.3961 
Quantile  0.5   22.5029 
Quantile  0.75  22.6118 
Quantile  0.975 23.0026 
Mean            24.4007 
Stdev           0.239866 
Quantile  0.025 23.8809 
Quantile  0.25  24.2824 
Quantile  0.5   24.3991 
Quantile  0.75  24.5162 
Quantile  0.975 24.9269 
Mean            19.9995 
Stdev           0.196835 
Quantile  0.025 19.5713 
Quantile  0.25  19.9028 
Quantile  0.5   19.9986 
Quantile  0.75  20.0945 
Quantile  0.975 20.4298 
Mean            21.7058 
Stdev           0.21356 
Quantile  0.025 21.251 
Quantile  0.25  21.5991 
Quantile  0.5   21.7022 
Quantile  0.75  21.8071 
Quantile  0.975 22.1822 
Mean            19.3037 
Stdev           0.189862 
Quantile  0.025 18.8972 
Quantile  0.25  19.2093 
Quantile  0.5   19.3012 
Quantile  0.75  19.3942 
Quantile  0.975 19.7251 
Mean            22.4029 
Stdev           0.220027 
Quantile  0.025 21.9296 
Quantile  0.25  22.2937 
Quantile  0.5   22.4005 
Quantile  0.75  22.5082 
Quantile  0.975 22.8889 
Mean            28.0961 
Stdev           0.276149 
Quantile  0.025 27.4905 
Quantile  0.25  27.9612 
Quantile  0.5   28.0962 
Quantile  0.75  28.2304 
Quantile  0.975 28.6948 
Mean            23.6946 
Stdev           0.233049 
Quantile  0.025 23.1803 
Quantile  0.25  23.5814 
Quantile  0.5   23.6955 
Quantile  0.75  23.8085 
Quantile  0.975 24.1967 
Mean            25.0071 
Stdev           0.246215 
Quantile  0.025 24.4834 
Quantile  0.25  24.8841 
Quantile  0.5   25.0028 
Quantile  0.75  25.1237 
Quantile  0.975 25.5569 
Mean            23.299 
Stdev           0.228372 
Quantile  0.025 22.8018 
Quantile  0.25  23.1866 
Quantile  0.5   23.2982 
Quantile  0.75  23.4097 
Quantile  0.975 23.7975 
Mean            28.711 
Stdev           0.282508 
Quantile  0.025 28.1146 
Quantile  0.25  28.569 
Quantile  0.5   28.7049 
Quantile  0.75  28.8441 
Quantile  0.975 29.3461 
Mean            21.5103 
Stdev           0.211681 
Quantile  0.025 21.0666 
Quantile  0.25  21.4033 
Quantile  0.5   21.5049 
Quantile  0.75  21.6094 
Quantile  0.975 21.9891 
Mean            23.0068 
Stdev           0.226683 
Quantile  0.025 22.525 
Quantile  0.25  22.8935 
Quantile  0.5   23.0027 
Quantile  0.75  23.114 
Quantile  0.975 23.5135 
Mean            26.7129 
Stdev           0.262885 
Quantile  0.025 26.1621 
Quantile  0.25  26.58 
Quantile  0.5   26.7061 
Quantile  0.75  26.836 
Quantile  0.975 27.3077 
Mean            21.7043 
Stdev           0.213892 
Quantile  0.025 21.2464 
Quantile  0.25  21.598 
Quantile  0.5   21.7013 
Quantile  0.75  21.8061 
Quantile  0.975 22.1792 
Mean            27.5087 
Stdev           0.270724 
Quantile  0.025 26.9342 
Quantile  0.25  27.3731 
Quantile  0.5   27.5036 
Quantile  0.75  27.6367 
Quantile  0.975 28.1145 
Mean            30.092 
Stdev           0.295989 
Quantile  0.025 29.4371 
Quantile  0.25  29.9485 
Quantile  0.5   30.0937 
Quantile  0.75  30.237 
Quantile  0.975 30.7279 
Mean            44.7912 
Stdev           0.440266 
Quantile  0.025 43.822 
Quantile  0.25  44.5769 
Quantile  0.5   44.7924 
Quantile  0.75  45.0061 
Quantile  0.975 45.742 
Mean            50.0078 
Stdev           0.491234 
Quantile  0.025 48.9531 
Quantile  0.25  49.7637 
Quantile  0.5   50.0019 
Quantile  0.75  50.2426 
Quantile  0.975 51.095 
Mean            37.6162 
Stdev           0.370479 
Quantile  0.025 36.8369 
Quantile  0.25  37.4296 
Quantile  0.5   37.6075 
Quantile  0.75  37.7901 
Quantile  0.975 38.4518 
Mean            31.6087 
Stdev           0.3112 
Quantile  0.025 30.9463 
Quantile  0.25  31.4532 
Quantile  0.5   31.6034 
Quantile  0.75  31.7562 
Quantile  0.975 32.3032 
Mean            46.6765 
Stdev           0.459464 
Quantile  0.025 45.6434 
Quantile  0.25  46.4568 
Quantile  0.5   46.6837 
Quantile  0.75  46.9047 
Quantile  0.975 47.6465 
Mean            31.4888 
Stdev           0.309407 
Quantile  0.025 30.8002 
Quantile  0.25  31.3395 
Quantile  0.5   31.4917 
Quantile  0.75  31.6413 
Quantile  0.975 32.1492 
Mean            24.3033 
Stdev           0.239138 
Quantile  0.025 23.7892 
Quantile  0.25  24.1848 
Quantile  0.5   24.3006 
Quantile  0.75  24.4176 
Quantile  0.975 24.8321 
Mean            31.7142 
Stdev           0.312391 
Quantile  0.025 31.058 
Quantile  0.25  31.5567 
Quantile  0.5   31.7067 
Quantile  0.75  31.8607 
Quantile  0.975 32.4197 
Mean            41.7181 
Stdev           0.4109 
Quantile  0.025 40.8539 
Quantile  0.25  41.5111 
Quantile  0.5   41.7084 
Quantile  0.75  41.911 
Quantile  0.975 42.645 
Mean            48.2703 
Stdev           0.47488 
Quantile  0.025 47.1949 
Quantile  0.25  48.0446 
Quantile  0.5   48.2799 
Quantile  0.75  48.5078 
Quantile  0.975 49.2645 
Mean            28.9953 
Stdev           0.284492 
Quantile  0.025 28.3707 
Quantile  0.25  28.8564 
Quantile  0.5   28.9957 
Quantile  0.75  29.1341 
Quantile  0.975 29.6111 
Mean            23.9981 
Stdev           0.236023 
Quantile  0.025 23.4828 
Quantile  0.25  23.8825 
Quantile  0.5   23.9976 
Quantile  0.75  24.1125 
Quantile  0.975 24.5121 
Mean            25.1123 
Stdev           0.247438 
Quantile  0.025 24.5941 
Quantile  0.25  24.9873 
Quantile  0.5   25.1059 
Quantile  0.75  25.228 
Quantile  0.975 25.6725 
Mean            31.5171 
Stdev           0.3105 
Quantile  0.025 30.8696 
Quantile  0.25  31.3598 
Quantile  0.5   31.5084 
Quantile  0.75  31.6619 
Quantile  0.975 32.2226 
Mean            23.7142 
Stdev           0.233887 
Quantile  0.025 23.2284 
Quantile  0.25  23.5954 
Quantile  0.5   23.7071 
Quantile  0.75  23.8228 
Quantile  0.975 24.2476 
Mean            23.3116 
Stdev           0.229858 
Quantile  0.025 22.8306 
Quantile  0.25  23.1955 
Quantile  0.5   23.3056 
Quantile  0.75  23.419 
Quantile  0.975 23.8324 
Mean            22.0337 
Stdev           0.220331 
Quantile  0.025 21.6086 
Quantile  0.25  21.9168 
Quantile  0.5   22.0186 
Quantile  0.75  22.1297 
Quantile  0.975 22.5647 
Mean            20.1067 
Stdev           0.19815 
Quantile  0.025 19.6867 
Quantile  0.25  20.0075 
Quantile  0.5   20.1028 
Quantile  0.75  20.2002 
Quantile  0.975 20.5507 
Mean            22.2045 
Stdev           0.218144 
Quantile  0.025 21.7379 
Quantile  0.25  22.0958 
Quantile  0.5   22.2015 
Quantile  0.75  22.3085 
Quantile  0.975 22.689 
Mean            23.7018 
Stdev           0.232747 
Quantile  0.025 23.1992 
Quantile  0.25  23.5867 
Quantile  0.5   23.6998 
Quantile  0.75  23.8136 
Quantile  0.975 24.2141 
Mean            17.6087 
Stdev           0.173624 
Quantile  0.025 17.2453 
Quantile  0.25  17.521 
Quantile  0.5   17.6042 
Quantile  0.75  17.6899 
Quantile  0.975 18.0021 
Mean            18.5065 
Stdev           0.182169 
Quantile  0.025 18.1209 
Quantile  0.25  18.4151 
Quantile  0.5   18.5028 
Quantile  0.75  18.5924 
Quantile  0.975 18.9151 
Mean            24.2898 
Stdev           0.238722 
Quantile  0.025 23.7562 
Quantile  0.25  24.175 
Quantile  0.5   24.2927 
Quantile  0.75  24.4079 
Quantile  0.975 24.7969 
Mean            20.5078 
Stdev           0.202034 
Quantile  0.025 20.0811 
Quantile  0.25  20.4063 
Quantile  0.5   20.5035 
Quantile  0.75  20.6029 
Quantile  0.975 20.9619 
Mean            24.5021 
Stdev           0.240897 
Quantile  0.025 23.9822 
Quantile  0.25  24.3829 
Quantile  0.5   24.4999 
Quantile  0.75  24.6177 
Quantile  0.975 25.0327 
Mean            26.2027 
Stdev           0.257476 
Quantile  0.025 25.6478 
Quantile  0.25  26.0752 
Quantile  0.5   26.2002 
Quantile  0.75  26.3262 
Quantile  0.975 26.7706 
Mean            24.4022 
Stdev           0.240069 
Quantile  0.025 23.8843 
Quantile  0.25  24.2835 
Quantile  0.5   24.4 
Quantile  0.75  24.5173 
Quantile  0.975 24.9313 
Mean            24.8067 
Stdev           0.244075 
Quantile  0.025 24.2871 
Quantile  0.25  24.6848 
Quantile  0.5   24.8026 
Quantile  0.75  24.9225 
Quantile  0.975 25.3513 
Mean            29.6005 
Stdev           0.291158 
Quantile  0.025 28.969 
Quantile  0.25  29.4571 
Quantile  0.5   29.5987 
Quantile  0.75  29.7407 
Quantile  0.975 30.2388 
Mean            42.793 
Stdev           0.420894 
Quantile  0.025 41.8685 
Quantile  0.25  42.5878 
Quantile  0.5   42.7936 
Quantile  0.75  42.998 
Quantile  0.975 43.7041 
Mean            21.9039 
Stdev           0.215696 
Quantile  0.025 21.4415 
Quantile  0.25  21.7968 
Quantile  0.5   21.9011 
Quantile  0.75  22.0067 
Quantile  0.975 22.3822 
Mean            20.9053 
Stdev           0.20593 
Quantile  0.025 20.4662 
Quantile  0.25  20.8026 
Quantile  0.5   20.902 
Quantile  0.75  21.003 
Quantile  0.975 21.3642 
Mean            43.9227 
Stdev           0.438553 
Quantile  0.025 42.8608 
Quantile  0.25  43.7287 
Quantile  0.5   43.9522 
Quantile  0.75  44.1553 
Quantile  0.975 44.7622 
Mean            49.9826 
Stdev           0.491492 
Quantile  0.025 48.8892 
Quantile  0.25  49.7453 
Quantile  0.5   49.987 
Quantile  0.75  50.2246 
Quantile  0.975 51.0323 
Mean            35.9923 
Stdev           0.353774 
Quantile  0.025 35.2124 
Quantile  0.25  35.8202 
Quantile  0.5   35.9935 
Quantile  0.75  36.1651 
Quantile  0.975 36.7551 
Mean            30.1035 
Stdev           0.296574 
Quantile  0.025 29.4647 
Quantile  0.25  29.9568 
Quantile  0.5   30.1004 
Quantile  0.75  30.2453 
Quantile  0.975 30.7583 
Mean            33.8061 
Stdev           0.33228 
Quantile  0.025 33.094 
Quantile  0.25  33.6409 
Quantile  0.5   33.8018 
Quantile  0.75  33.9647 
Quantile  0.975 34.5428 
Mean            43.0846 
Stdev           0.424142 
Quantile  0.025 42.1402 
Quantile  0.25  42.8802 
Quantile  0.5   43.0886 
Quantile  0.75  43.2934 
Quantile  0.975 43.9898 
Mean            48.7937 
Stdev           0.479596 
Quantile  0.025 47.7429 
Quantile  0.25  48.5593 
Quantile  0.5   48.7937 
Quantile  0.75  49.0269 
Quantile  0.975 49.8345 
Mean            31.0216 
Stdev           0.30626 
Quantile  0.025 30.3903 
Quantile  0.25  30.8653 
Quantile  0.5   31.011 
Quantile  0.75  31.1629 
Quantile  0.975 31.7244 
Mean            36.4813 
Stdev           0.358684 
Quantile  0.025 35.6744 
Quantile  0.25  36.3097 
Quantile  0.5   36.487 
Quantile  0.75  36.6597 
Quantile  0.975 37.2379 
Mean            22.7901 
Stdev           0.224276 
Quantile  0.025 22.2881 
Quantile  0.25  22.6824 
Quantile  0.5   22.7929 
Quantile  0.75  22.901 
Quantile  0.975 23.2659 
Mean            30.705 
Stdev           0.30178 
Quantile  0.025 30.0574 
Quantile  0.25  30.5551 
Quantile  0.5   30.7013 
Quantile  0.75  30.8492 
Quantile  0.975 31.3732 
Mean            49.9784 
Stdev           0.491226 
Quantile  0.025 48.8793 
Quantile  0.25  49.7423 
Quantile  0.5   49.9846 
Quantile  0.75  50.2215 
Quantile  0.975 51.0208 
Mean            43.4718 
Stdev           0.428007 
Quantile  0.025 42.5003 
Quantile  0.25  43.2688 
Quantile  0.5   43.4811 
Quantile  0.75  43.6862 
Quantile  0.975 44.3656 
Mean            20.708 
Stdev           0.203767 
Quantile  0.025 20.278 
Quantile  0.25  20.6056 
Quantile  0.5   20.7036 
Quantile  0.75  20.804 
Quantile  0.975 21.1662 
Mean            21.1002 
Stdev           0.207544 
Quantile  0.025 20.6498 
Quantile  0.25  20.998 
Quantile  0.5   21.099 
Quantile  0.75  21.2002 
Quantile  0.975 21.555 
Mean            25.1982 
Stdev           0.247671 
Quantile  0.025 24.6576 
Quantile  0.25  25.0767 
Quantile  0.5   25.1976 
Quantile  0.75  25.3181 
Quantile  0.975 25.7377 
Mean            24.4064 
Stdev           0.240272 
Quantile  0.025 23.8944 
Quantile  0.25  24.2865 
Quantile  0.5   24.4024 
Quantile  0.75  24.5203 
Quantile  0.975 24.9421 
Mean            35.2087 
Stdev           0.346389 
Quantile  0.025 34.47 
Quantile  0.25  35.0359 
Quantile  0.5   35.2032 
Quantile  0.75  35.3732 
Quantile  0.975 35.9804 
Mean            32.3883 
Stdev           0.318249 
Quantile  0.025 31.6798 
Quantile  0.25  32.2348 
Quantile  0.5   32.3914 
Quantile  0.75  32.5452 
Quantile  0.975 33.0673 
Mean            32.002 
Stdev           0.315016 
Quantile  0.025 31.321 
Quantile  0.25  31.8465 
Quantile  0.5   31.9994 
Quantile  0.75  32.1532 
Quantile  0.975 32.6949 
Mean            33.2137 
Stdev           0.327071 
Quantile  0.025 32.5246 
Quantile  0.25  33.0491 
Quantile  0.5   33.2063 
Quantile  0.75  33.3674 
Quantile  0.975 33.9504 
Mean            33.1039 
Stdev           0.325306 
Quantile  0.025 32.4035 
Quantile  0.25  32.9427 
Quantile  0.5   33.1005 
Quantile  0.75  33.2597 
Quantile  0.975 33.822 
Mean            29.1059 
Stdev           0.286112 
Quantile  0.025 28.4938 
Quantile  0.25  28.9635 
Quantile  0.5   29.1019 
Quantile  0.75  29.2423 
Quantile  0.975 29.7413 
Mean            35.0898 
Stdev           0.345157 
Quantile  0.025 34.325 
Quantile  0.25  34.9227 
Quantile  0.5   35.0921 
Quantile  0.75  35.2592 
Quantile  0.975 35.8302 
Mean            45.3864 
Stdev           0.445648 
Quantile  0.025 44.3983 
Quantile  0.25  45.1705 
Quantile  0.5   45.3896 
Quantile  0.75  45.6054 
Quantile  0.975 46.3413 
Mean            35.3827 
Stdev           0.348052 
Quantile  0.025 34.6008 
Quantile  0.25  35.216 
Quantile  0.5   35.3879 
Quantile  0.75  35.5555 
Quantile  0.975 36.1181 
Mean            45.9676 
Stdev           0.452762 
Quantile  0.025 44.9362 
Quantile  0.25  45.7536 
Quantile  0.5   45.9785 
Quantile  0.75  46.1951 
Quantile  0.975 46.9091 
Mean            49.9462 
Stdev           0.492939 
Quantile  0.025 48.7972 
Quantile  0.25  49.7182 
Quantile  0.5   49.9657 
Quantile  0.75  50.1992 
Quantile  0.975 50.9421 
Mean            32.1974 
Stdev           0.316662 
Quantile  0.025 31.5057 
Quantile  0.25  32.0423 
Quantile  0.5   32.1967 
Quantile  0.75  32.3508 
Quantile  0.975 32.8868 
Mean            22.0095 
Stdev           0.217194 
Quantile  0.025 21.5526 
Quantile  0.25  21.9002 
Quantile  0.5   22.0044 
Quantile  0.75  22.1113 
Quantile  0.975 22.4995 
Mean            20.1085 
Stdev           0.198288 
Quantile  0.025 19.6911 
Quantile  0.25  20.0087 
Quantile  0.5   20.1039 
Quantile  0.75  20.2016 
Quantile  0.975 20.5555 
Mean            23.197 
Stdev           0.228287 
Quantile  0.025 22.6967 
Quantile  0.25  23.0855 
Quantile  0.5   23.197 
Quantile  0.75  23.3079 
Quantile  0.975 23.6925 
Mean            22.3053 
Stdev           0.219564 
Quantile  0.025 21.8367 
Quantile  0.25  22.1959 
Quantile  0.5   22.3019 
Quantile  0.75  22.4096 
Quantile  0.975 22.7941 
Mean            24.8084 
Stdev           0.244035 
Quantile  0.025 24.2914 
Quantile  0.25  24.686 
Quantile  0.5   24.8036 
Quantile  0.75  24.9237 
Quantile  0.975 25.3553 
Mean            28.5093 
Stdev           0.280418 
Quantile  0.025 27.9147 
Quantile  0.25  28.3687 
Quantile  0.5   28.5039 
Quantile  0.75  28.6419 
Quantile  0.975 29.1371 
Mean            37.2871 
Stdev           0.366818 
Quantile  0.025 36.4711 
Quantile  0.25  37.1101 
Quantile  0.5   37.2904 
Quantile  0.75  37.4676 
Quantile  0.975 38.0706 
Mean            27.8967 
Stdev           0.27371 
Quantile  0.025 27.2975 
Quantile  0.25  27.7627 
Quantile  0.5   27.8966 
Quantile  0.75  28.0298 
Quantile  0.975 28.4909 
Mean            23.8965 
Stdev           0.234731 
Quantile  0.025 23.3816 
Quantile  0.25  23.7819 
Quantile  0.5   23.8967 
Quantile  0.75  24.0108 
Quantile  0.975 24.4052 
Mean            21.7064 
Stdev           0.213731 
Quantile  0.025 21.2521 
Quantile  0.25  21.5995 
Quantile  0.5   21.7026 
Quantile  0.75  21.8075 
Quantile  0.975 22.184 
Mean            28.5983 
Stdev           0.281643 
Quantile  0.025 27.9841 
Quantile  0.25  28.4603 
Quantile  0.5   28.5975 
Quantile  0.75  28.7345 
Quantile  0.975 29.2127 
Mean            27.0947 
Stdev           0.266152 
Quantile  0.025 26.5089 
Quantile  0.25  26.9651 
Quantile  0.5   27.0954 
Quantile  0.75  27.2247 
Quantile  0.975 27.6694 
Mean            20.3021 
Stdev           0.19987 
Quantile  0.025 19.8712 
Quantile  0.25  20.2032 
Quantile  0.5   20.3001 
Quantile  0.75  20.3978 
Quantile  0.975 20.743 
Mean            22.5038 
Stdev           0.221179 
Quantile  0.025 22.0294 
Quantile  0.25  22.3939 
Quantile  0.5   22.501 
Quantile  0.75  22.6094 
Quantile  0.975 22.9938 
Mean            28.998 
Stdev           0.285779 
Quantile  0.025 28.3743 
Quantile  0.25  28.8581 
Quantile  0.5   28.9973 
Quantile  0.75  29.1362 
Quantile  0.975 29.6209 
Mean            24.8034 
Stdev           0.244378 
Quantile  0.025 24.278 
Quantile  0.25  24.6824 
Quantile  0.5   24.8007 
Quantile  0.75  24.9201 
Quantile  0.975 25.3439 
Mean            22.0156 
Stdev           0.217377 
Quantile  0.025 21.568 
Quantile  0.25  21.9046 
Quantile  0.5   22.008 
Quantile  0.75  22.1158 
Quantile  0.975 22.5148 
Mean            26.3945 
Stdev           0.259763 
Quantile  0.025 25.822 
Quantile  0.25  26.2682 
Quantile  0.5   26.3953 
Quantile  0.75  26.5212 
Quantile  0.975 26.955 
Mean            33.0796 
Stdev           0.325243 
Quantile  0.025 32.3431 
Quantile  0.25  32.925 
Quantile  0.5   33.0862 
Quantile  0.75  33.2424 
Quantile  0.975 33.7604 
Mean            36.0749 
Stdev           0.354885 
Quantile  0.025 35.2671 
Quantile  0.25  35.907 
Quantile  0.5   36.0833 
Quantile  0.75  36.2533 
Quantile  0.975 36.8133 
Mean            28.3907 
Stdev           0.279111 
Quantile  0.025 27.7707 
Quantile  0.25  28.2558 
Quantile  0.5   28.393 
Quantile  0.75  28.528 
Quantile  0.975 28.9877 
Mean            33.3961 
Stdev           0.328249 
Quantile  0.025 32.6774 
Quantile  0.25  33.2356 
Quantile  0.5   33.3959 
Quantile  0.75  33.5556 
Quantile  0.975 34.1089 
Mean            28.1907 
Stdev           0.276822 
Quantile  0.025 27.5757 
Quantile  0.25  28.0568 
Quantile  0.5   28.193 
Quantile  0.75  28.327 
Quantile  0.975 28.7825 
Mean            22.8072 
Stdev           0.224457 
Quantile  0.025 22.331 
Quantile  0.25  22.6948 
Quantile  0.5   22.803 
Quantile  0.75  22.9134 
Quantile  0.975 23.3096 
Mean            20.2974 
Stdev           0.199378 
Quantile  0.025 19.8607 
Quantile  0.25  20.1999 
Quantile  0.5   20.2974 
Quantile  0.75  20.3944 
Quantile  0.975 20.7301 
Mean            16.0907 
Stdev           0.15836 
Quantile  0.025 15.733 
Quantile  0.25  16.0153 
Quantile  0.5   16.0937 
Quantile  0.75  16.1697 
Quantile  0.975 16.4232 
Mean            22.0983 
Stdev           0.217627 
Quantile  0.025 21.6231 
Quantile  0.25  21.9918 
Quantile  0.5   22.0978 
Quantile  0.75  22.2036 
Quantile  0.975 22.5724 
Mean            19.4047 
Stdev           0.191138 
Quantile  0.025 18.9968 
Quantile  0.25  19.3094 
Quantile  0.5   19.4017 
Quantile  0.75  19.4954 
Quantile  0.975 19.8303 
Mean            21.6 
Stdev           0.212196 
Quantile  0.025 21.1393 
Quantile  0.25  21.4955 
Quantile  0.5   21.5988 
Quantile  0.75  21.7024 
Quantile  0.975 22.0646 
Mean            23.7967 
Stdev           0.233891 
Quantile  0.025 23.2837 
Quantile  0.25  23.6824 
Quantile  0.5   23.7968 
Quantile  0.75  23.9104 
Quantile  0.975 24.3037 
Mean            16.2052 
Stdev           0.159694 
Quantile  0.025 15.8665 
Quantile  0.25  16.1253 
Quantile  0.5   16.2022 
Quantile  0.75  16.2807 
Quantile  0.975 16.5629 
Mean            17.8053 
Stdev           0.17521 
Quantile  0.025 17.433 
Quantile  0.25  17.7176 
Quantile  0.5   17.8021 
Quantile  0.75  17.8882 
Quantile  0.975 18.1969 
Mean            19.7981 
Stdev           0.194363 
Quantile  0.025 19.3733 
Quantile  0.25  19.7029 
Quantile  0.5   19.7978 
Quantile  0.75  19.8925 
Quantile  0.975 20.2208 
Mean            23.0989 
Stdev           0.227184 
Quantile  0.025 22.604 
Quantile  0.25  22.9874 
Quantile  0.5   23.0981 
Quantile  0.75  23.2088 
Quantile  0.975 23.5948 
Mean            21.0062 
Stdev           0.206346 
Quantile  0.025 20.5677 
Quantile  0.25  20.9029 
Quantile  0.5   21.0025 
Quantile  0.75  21.104 
Quantile  0.975 21.4672 
Mean            23.8004 
Stdev           0.23426 
Quantile  0.025 23.2924 
Quantile  0.25  23.6851 
Quantile  0.5   23.799 
Quantile  0.75  23.9132 
Quantile  0.975 24.3141 
Mean            23.1023 
Stdev           0.227285 
Quantile  0.025 22.6122 
Quantile  0.25  22.9898 
Quantile  0.5   23.1001 
Quantile  0.75  23.2112 
Quantile  0.975 23.6034 
Mean            20.4031 
Stdev           0.200898 
Quantile  0.025 19.9717 
Quantile  0.25  20.3035 
Quantile  0.5   20.4007 
Quantile  0.75  20.499 
Quantile  0.975 20.8478 
Mean            18.5017 
Stdev           0.181798 
Quantile  0.025 18.1096 
Quantile  0.25  18.4117 
Quantile  0.5   18.5 
Quantile  0.75  18.5889 
Quantile  0.975 18.9023 
Mean            24.9919 
Stdev           0.245406 
Quantile  0.025 24.447 
Quantile  0.25  24.8732 
Quantile  0.5   24.9939 
Quantile  0.75  25.1127 
Quantile  0.975 25.5169 
Mean            24.5937 
Stdev           0.241603 
Quantile  0.025 24.0595 
Quantile  0.25  24.4764 
Quantile  0.5   24.595 
Quantile  0.75  24.7121 
Quantile  0.975 25.113 
Mean            23.0003 
Stdev           0.226234 
Quantile  0.025 22.5094 
Quantile  0.25  22.8889 
Quantile  0.5   22.9989 
Quantile  0.75  23.1093 
Quantile  0.975 23.4961 
Mean            22.1984 
Stdev           0.218187 
Quantile  0.025 21.7223 
Quantile  0.25  22.0914 
Quantile  0.5   22.1979 
Quantile  0.75  22.3041 
Quantile  0.975 22.6738 
Mean            19.3033 
Stdev           0.189958 
Quantile  0.025 18.8959 
Quantile  0.25  19.2089 
Quantile  0.5   19.3009 
Quantile  0.75  19.3939 
Quantile  0.975 19.7243 
Mean            22.6068 
Stdev           0.222465 
Quantile  0.025 22.1343 
Quantile  0.25  22.4955 
Quantile  0.5   22.6028 
Quantile  0.75  22.7121 
Quantile  0.975 23.1042 
Mean            19.8078 
Stdev           0.194915 
Quantile  0.025 19.3966 
Quantile  0.25  19.7098 
Quantile  0.5   19.8035 
Quantile  0.75  19.8996 
Quantile  0.975 20.2463 
Mean            17.1019 
Stdev           0.168604 
Quantile  0.025 16.7388 
Quantile  0.25  17.0186 
Quantile  0.5   17.1002 
Quantile  0.75  17.1826 
Quantile  0.975 17.4742 
Mean            19.4029 
Stdev           0.191047 
Quantile  0.025 18.9925 
Quantile  0.25  19.3082 
Quantile  0.5   19.4007 
Quantile  0.75  19.4941 
Quantile  0.975 19.8257 
Mean            22.2007 
Stdev           0.21852 
Quantile  0.025 21.7272 
Quantile  0.25  22.093 
Quantile  0.5   22.1992 
Quantile  0.75  22.3058 
Quantile  0.975 22.6803 
Mean            20.7055 
Stdev           0.203974 
Quantile  0.025 20.2709 
Quantile  0.25  20.6037 
Quantile  0.5   20.7021 
Quantile  0.75  20.8022 
Quantile  0.975 21.1604 
Mean            21.1005 
Stdev           0.207425 
Quantile  0.025 20.651 
Quantile  0.25  20.9983 
Quantile  0.5   21.0992 
Quantile  0.75  21.2004 
Quantile  0.975 21.5556 
Mean            19.503 
Stdev           0.19179 
Quantile  0.025 19.0911 
Quantile  0.25  19.4078 
Quantile  0.5   19.5007 
Quantile  0.75  19.5946 
Quantile  0.975 19.9274 
Mean            18.5065 
Stdev           0.182403 
Quantile  0.025 18.1205 
Quantile  0.25  18.4151 
Quantile  0.5   18.5028 
Quantile  0.75  18.5925 
Quantile  0.975 18.9158 
Mean            20.6025 
Stdev           0.202458 
Quantile  0.025 20.1668 
Quantile  0.25  20.5021 
Quantile  0.5   20.6004 
Quantile  0.75  20.6995 
Quantile  0.975 21.0496 
Mean            19.0022 
Stdev           0.18696 
Quantile  0.025 18.5997 
Quantile  0.25  18.9097 
Quantile  0.5   19.0003 
Quantile  0.75  19.0917 
Quantile  0.975 19.4151 
Mean            18.6977 
Stdev           0.183889 
Quantile  0.025 18.295 
Quantile  0.25  18.6078 
Quantile  0.5   18.6977 
Quantile  0.75  18.787 
Quantile  0.975 19.097 
Mean            32.6854 
Stdev           0.321868 
Quantile  0.025 31.9645 
Quantile  0.25  32.5311 
Quantile  0.5   32.6897 
Quantile  0.75  32.8447 
Quantile  0.975 33.3679 
Mean            16.5129 
Stdev           0.163492 
Quantile  0.025 16.1779 
Quantile  0.25  16.4292 
Quantile  0.5   16.5067 
Quantile  0.75  16.5878 
Quantile  0.975 16.8901 
Mean            23.9005 
Stdev           0.234947 
Quantile  0.025 23.3912 
Quantile  0.25  23.7847 
Quantile  0.5   23.899 
Quantile  0.75  24.0137 
Quantile  0.975 24.4158 
Mean            31.1673 
Stdev           0.307493 
Quantile  0.025 30.4517 
Quantile  0.25  31.0248 
Quantile  0.5   31.1791 
Quantile  0.75  31.3248 
Quantile  0.975 31.7898 
Mean            17.503 
Stdev           0.172591 
Quantile  0.025 17.1327 
Quantile  0.25  17.4174 
Quantile  0.5   17.5008 
Quantile  0.75  17.5852 
Quantile  0.975 17.8855 
Mean            17.2083 
Stdev           0.169656 
Quantile  0.025 16.8528 
Quantile  0.25  17.1227 
Quantile  0.5   17.204 
Quantile  0.75  17.2877 
Quantile  0.975 17.5923 
Mean            23.0975 
Stdev           0.227309 
Quantile  0.025 22.6001 
Quantile  0.25  22.9864 
Quantile  0.5   23.0973 
Quantile  0.75  23.2078 
Quantile  0.975 23.5915 
Mean            24.4999 
Stdev           0.240682 
Quantile  0.025 23.9771 
Quantile  0.25  24.3814 
Quantile  0.5   24.4986 
Quantile  0.75  24.616 
Quantile  0.975 25.0267 
Mean            26.6052 
Stdev           0.262009 
Quantile  0.025 26.0441 
Quantile  0.25  26.4749 
Quantile  0.5   26.6016 
Quantile  0.75  26.7299 
Quantile  0.975 27.1868 
Mean            22.902 
Stdev           0.225454 
Quantile  0.025 22.4155 
Quantile  0.25  22.7906 
Quantile  0.5   22.8999 
Quantile  0.75  23.0101 
Quantile  0.975 23.3988 
Mean            24.1018 
Stdev           0.237104 
Quantile  0.025 23.5897 
Quantile  0.25  23.9846 
Quantile  0.5   24.0998 
Quantile  0.75  24.2156 
Quantile  0.975 24.6237 
Mean            18.6057 
Stdev           0.183453 
Quantile  0.025 18.216 
Quantile  0.25  18.514 
Quantile  0.5   18.6023 
Quantile  0.75  18.6924 
Quantile  0.975 19.016 
Mean            30.0853 
Stdev           0.296318 
Quantile  0.025 29.4198 
Quantile  0.25  29.9436 
Quantile  0.5   30.0898 
Quantile  0.75  30.2323 
Quantile  0.975 30.7118 
Mean            18.2052 
Stdev           0.179249 
Quantile  0.025 17.8239 
Quantile  0.25  18.1156 
Quantile  0.5   18.2021 
Quantile  0.75  18.2901 
Quantile  0.975 18.6055 
Mean            20.6026 
Stdev           0.203125 
Quantile  0.025 20.1655 
Quantile  0.25  20.5021 
Quantile  0.5   20.6004 
Quantile  0.75  20.6997 
Quantile  0.975 21.0516 
# Posterior mean and 95% credible interval
map$PMoriginal <- sapply(marginals_summaries, '[[', "mean")
map$LLoriginal <- sapply(marginals_summaries, '[[', "quant0.025")
map$ULoriginal <- sapply(marginals_summaries, '[[', "quant0.975")
# Common legend
at <- seq(
  min(c(map$PMoriginal, map$LLoriginal, map$ULoriginal)),
  max(c(map$PMoriginal, map$LLoriginal, map$ULoriginal)),
  length.out = 8
)

# Popup table
popuptable <- leafpop::popupTable(
  dplyr::mutate_if(map, is.numeric, round, digits = 2),
  zcol = c("TOWN", "vble", "CRIM", "RM", "PM", "LL", "UL"),
  row.numbers = FALSE,
  feature.id = FALSE
)

# Map visualizations
m1 <- mapview(
  map,
  zcol = "PMoriginal",
  map.types = "CartoDB.Positron",
  at = at,
  popup = popuptable
)

m2 <- mapview(
  map,
  zcol = "LLoriginal",
  map.types = "CartoDB.Positron",
  at = at,
  popup = popuptable
)

m3 <- mapview(
  map,
  zcol = "ULoriginal",
  map.types = "CartoDB.Positron",
  at = at,
  popup = popuptable
)
m <- leafsync::sync(m1, m2, m3, ncol = 3)
m

Posterior mean of the housing prices (left), together with lower (center) and upper (right) limits of 95% credible intervals.

10 Disease risk modeling

10.1 Spatial disease risk models

Spatial disease risk models are commonly specified using a Poisson distribution for the observed number of cases (Yi) with mean equal to the expected number of cases (Ei) times the relative risk (??i) corresponding to area i, i = 1,…,n,

10.2 Modeling of lung cancer risk in Pennsylvania

library(SpatialEpi)
Warning: package 'SpatialEpi' was built under R version 4.4.3
Loading required package: sp
data(pennLC)
class(pennLC)
[1] "list"
names(pennLC)
[1] "geo"             "data"            "smoking"         "spatial.polygon"
head(pennLC$data)
  county cases population race gender      age
1  adams     0       1492    o      f Under.40
2  adams     0        365    o      f    40.59
3  adams     1         68    o      f    60.69
4  adams     0         73    o      f      70+
5  adams     0      23351    w      f Under.40
6  adams     5      12136    w      f    40.59
head(pennLC$smoking)
     county smoking
1     adams   0.234
2 allegheny   0.245
3 armstrong   0.250
4    beaver   0.276
5   bedford   0.228
6     berks   0.249
library(sf)

map <- st_as_sf(pennLC$spatial.polygon)

countynames <- sapply(
  slot(pennLC$spatial.polygon, "polygons"),
  function(x) { slot(x, "ID") }
)

map$county <- countynames

head(map)
Simple feature collection with 6 features and 1 field
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: -80.51776 ymin: 39.72889 xmax: -75.53303 ymax: 41.1441
Geodetic CRS:  +proj=longlat
                        geometry    county
1 POLYGON ((-77.4467 39.96954...     adams
2 POLYGON ((-80.14534 40.6742... allegheny
3 POLYGON ((-79.21142 40.9091... armstrong
4 POLYGON ((-80.1568 40.85189...    beaver
5 POLYGON ((-78.38063 39.7288...   bedford
6 POLYGON ((-75.53303 40.4508...     berks

Observed cases

library(dplyr)

Attaching package: 'dplyr'
The following objects are masked from 'package:stats':

    filter, lag
The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union
d <- group_by(pennLC$data, county) %>% summarize(Y= sum(cases))
head(d)
# A tibble: 6 × 2
  county        Y
  <fct>     <int>
1 adams        55
2 allegheny  1275
3 armstrong    49
4 beaver      172
5 bedford      37
6 berks       308

Expected cases

pennLC$data <- pennLC$data[order(pennLC$data$county,
pennLC$data$race, pennLC$data$gender, pennLC$data$age), ]
E <- expected(population = pennLC$data$population,
 cases = pennLC$data$cases, n.strata = 16)
d$E <- E
head(d)
# A tibble: 6 × 3
  county        Y      E
  <fct>     <int>  <dbl>
1 adams        55   69.6
2 allegheny  1275 1182. 
3 armstrong    49   67.6
4 beaver      172  173. 
5 bedford      37   44.2
6 berks       308  301. 

Smokers proportions

d <- dplyr::left_join(d, pennLC$smoking, by = "county")

Standardized Mortality Ratios

d$SMR <- d$Y/d$E
head(d)
# A tibble: 6 × 5
  county        Y      E smoking   SMR
  <fct>     <int>  <dbl>   <dbl> <dbl>
1 adams        55   69.6   0.234 0.790
2 allegheny  1275 1182.    0.245 1.08 
3 armstrong    49   67.6   0.25  0.725
4 beaver      172  173.    0.276 0.997
5 bedford      37   44.2   0.228 0.837
6 berks       308  301.    0.249 1.02 
map <- dplyr::left_join(map, d, by = "county")
library(mapview)
library(RColorBrewer)

pal <- colorRampPalette(brewer.pal(9, "YlOrRd"))

mapview(
  map,
  zcol = "SMR",
  color = "gray",
  alpha.regions = 0.8,
  layer.name = "SMR",
  col.regions = pal,
  map.types = "CartoDB.Positron"
)

Relative risks of the counties of Pennsylvania, USA.

library(mapview)
library(RColorBrewer)
library(leafpop)

pal <- colorRampPalette(brewer.pal(9, "YlOrRd"))
mapviewOptions(fgb = FALSE)

popuptable <- leafpop::popupTable(
  dplyr::mutate_if(map, is.numeric, round, digits = 2),
  zcol = c("county", "Y", "E", "smoking", "SMR"),
  row.numbers = FALSE,
  feature.id = FALSE
)

mapview(
  map,
  zcol = "SMR",
  color = "gray",
  col.regions = pal,
  highlight = leaflet::highlightOptions(weight = 4),
  popup = popuptable
)
library(spdep)
library(INLA)
nb <- poly2nb(map)
nb2INLA("map.adj", nb)
g <- inla.read.graph(filename = "map.adj")
map$re_u <- 1:nrow(map)
map$re_v <- 1:nrow(map)
formula <- Y ~ smoking + f(re_u, model = "besag", graph = g, scale.model = TRUE) + f(re_v, model = "iid")
res <- inla(formula, family = "poisson", data = map, E= E, control.predictor = list(compute = TRUE), control.compute = list(return.marginals.predictor = TRUE))
res$summary.fixed
                  mean        sd  0.025quant   0.5quant  0.975quant       mode
(Intercept) -0.3235144 0.1499851 -0.61968472 -0.3233618 -0.02842592 -0.3234193
smoking      1.1545904 0.6234222 -0.07736443  1.1559465  2.38003002  1.1562136
                     kld
(Intercept) 3.572898e-08
smoking     3.580630e-08

We see the intercept ??0 =-0.323 with a 95% credible interval equal to (-0.619,-0.029), and the coefficient of smoking is ??1 = 1.155 with a 95% credible interval equal to (-0.076, 2.378) This indicates a non-significant effect of smoking.

res$summary.fitted.values[1:3,]
                         mean         sd 0.025quant  0.5quant 0.975quant
fitted.Predictor.01 0.8779225 0.05818241  0.7644309 0.8776429  0.9936634
fitted.Predictor.02 1.0597610 0.02751494  1.0072677 1.0592514  1.1151485
fitted.Predictor.03 0.9644514 0.05103596  0.8598242 0.9656337  1.0622242
                         mode
fitted.Predictor.01 0.8776984
fitted.Predictor.02 1.0582241
fitted.Predictor.03 0.9680030
# Relative risk
map$RR <- res$summary.fitted.values[, "mean"]

# Lower and upper limits of 95% credible intervals
map$LL <- res$summary.fitted.values[, "0.025quant"]
map$UL <- res$summary.fitted.values[, "0.975quant"]
library(mapview)
library(RColorBrewer)
library(leafpop)

pal <- colorRampPalette(brewer.pal(9, "YlOrRd"))
mapviewOptions(fgb = FALSE)

mapview(
  map,
  zcol = "RR",
  color = "gray",
  col.regions = pal,
  highlight = leaflet::highlightOptions(weight = 4),
  popup = leafpop::popupTable(
    dplyr::mutate_if(map, is.numeric, round, digits = 2),
    zcol = c("county", "Y", "E", "smoking", "SMR", "RR", "LL", "UL"),
    row.numbers = FALSE,
    feature.id = FALSE
  )
)

Relative risks of the counties of Pennsylvania, USA.

Comparing SMR and RR maps

at <- seq(min(map$SMR), max(map$SMR), length.out = 8)

m1 <- mapview(
  map,
  zcol = "SMR",
  color = "gray",
  col.regions = pal,
  at = at
)

m2 <- mapview(
  map,
  zcol = "RR",
  color = "gray",
  col.regions = pal,
  at = at
)

leafsync::sync(m1, m2)

SMRs (left) and RRs (right) of the counties of Pennsylvania, USA.

Exceedance probabilities

c<-1.2
marg<-res$marginals.fitted.values[[51]]
1-inla.pmarginal(q = c,marginal= marg)
[1] 0.05636408
library(ggplot2)

marginal <- inla.smarginal(res$marginals.fitted.values[[51]])
marginal <- data.frame(marginal)

ggplot(marginal, aes(x = x, y = y)) +
  geom_line() +
  labs(x = expression(theta[51]), y = "Density") +
  geom_vline(xintercept = 1.2, col = "black") +
  theme_bw(base_size = 20)

Posterior distribution of the relative risk for area 51 exceeding the threshold value of 1.2. The vertical line indicates the threshold value.

c<-1.2
map$exc<-sapply(res$marginals.fitted.values,
FUN= function(marg){1-inla.pmarginal(q=c,marginal= marg)})
pal <- colorRampPalette(brewer.pal(9, "YlOrRd"))
mapview(map, zcol = "exc", color = "gray", col.regions = pal, map.types = "CartoDB.Positron")

Probabilities that the relative risks of counties exceed 1.2.

11 Areal data issues

Spatial analyses of aggregated data often face the Misaligned Data Problem (MIDP), which occurs when the scale of the analyzed data differs from the scale at which it was originally collected. This misalignment can result in a loss of spatial detail, potentially obscuring important patterns and leading to biased or misleading conclusions (Banerjee et al., 2004).

Another common issue is the Modifiable Areal Unit Problem (MAUP) (Openshaw, 1984), where results vary depending on the level or configuration of spatial aggregation. The MAUP consists of two effects: the scale effect, where results change with the level of aggregation, and the zoning effect, where arbitrary boundary definitions influence outcomes.

Ecological studies (Robinson, 1950) analyze relationships between exposures and outcomes at the group level rather than the individual level. While useful when individual data are unavailable, they are prone to the ecological fallacy-group-level associations that do not necessarily apply to individuals. This introduces ecological bias, a specific form of MAUP that includes aggregation and specification biases (Gotway & Young, 2002).

Finally, integrating spatial data from different sources or resolutions-such as monitoring stations and satellite imagery-can improve prediction accuracy. Moraga et al. (2017) proposed a Bayesian melding model combining spatially misaligned data using INLA and SPDE for efficient inference. Zhong and Moraga (2023) compared this model with a Bayesian downscaler, demonstrating its ability to disaggregate areal data and produce spatially continuous predictions that enhance policy-relevant decision-making.